In[163]:= a[n_] := Binomial[2n,n] + 2^n + n^3 ; 

In[164]:= Table[a[n], {n, 0, 10}]

Out[164]= {2, 5, 18, 55, 150, 409, 1204, 3903, 13638, 49861, 186780}

In[165]:= ansatz = Sum[c[i,j]n^j f[n+i], {i, 0, 3}, {j, 0, 10}];

In[166]:= ansatz

                                           2                 3                 4
Out[166]= c[0, 0] f[n] + n c[0, 1] f[n] + n  c[0, 2] f[n] + n  c[0, 3] f[n] + n  c[0, 4] f[n] + 
 
      5                 6                 7                 8                 9
>    n  c[0, 5] f[n] + n  c[0, 6] f[n] + n  c[0, 7] f[n] + n  c[0, 8] f[n] + n  c[0, 9] f[n] + 
 
      10                                                          2
>    n   c[0, 10] f[n] + c[1, 0] f[1 + n] + n c[1, 1] f[1 + n] + n  c[1, 2] f[1 + n] + 
 
      3                     4                     5                     6
>    n  c[1, 3] f[1 + n] + n  c[1, 4] f[1 + n] + n  c[1, 5] f[1 + n] + n  c[1, 6] f[1 + n] + 
 
      7                     8                     9                     10
>    n  c[1, 7] f[1 + n] + n  c[1, 8] f[1 + n] + n  c[1, 9] f[1 + n] + n   c[1, 10] f[1 + n] + 
 
                                              2                     3
>    c[2, 0] f[2 + n] + n c[2, 1] f[2 + n] + n  c[2, 2] f[2 + n] + n  c[2, 3] f[2 + n] + 
 
      4                     5                     6                     7
>    n  c[2, 4] f[2 + n] + n  c[2, 5] f[2 + n] + n  c[2, 6] f[2 + n] + n  c[2, 7] f[2 + n] + 
 
      8                     9                     10
>    n  c[2, 8] f[2 + n] + n  c[2, 9] f[2 + n] + n   c[2, 10] f[2 + n] + c[3, 0] f[3 + n] + 
 
                           2                     3                     4
>    n c[3, 1] f[3 + n] + n  c[3, 2] f[3 + n] + n  c[3, 3] f[3 + n] + n  c[3, 4] f[3 + n] + 
 
      5                     6                     7                     8
>    n  c[3, 5] f[3 + n] + n  c[3, 6] f[3 + n] + n  c[3, 7] f[3 + n] + n  c[3, 8] f[3 + n] + 
 
      9                     10
>    n  c[3, 9] f[3 + n] + n   c[3, 10] f[3 + n]

In[167]:= sys = Table[ansatz, {n, 0, 50}] /. f -> a

Out[167]= {2 c[0, 0] + 5 c[1, 0] + 18 c[2, 0] + 55 c[3, 0], 
 
>    5 c[0, 0] + 5 c[0, 1] + 5 c[0, 2] + 5 c[0, 3] + 5 c[0, 4] + 5 c[0, 5] + 5 c[0, 6] + 5 c[0, 7] + 
 
>     5 c[0, 8] + 5 c[0, 9] + 5 c[0, 10] + 18 c[1, 0] + 18 c[1, 1] + 18 c[1, 2] + 18 c[1, 3] + 
 
>     18 c[1, 4] + 18 c[1, 5] + 18 c[1, 6] + 18 c[1, 7] + 18 c[1, 8] + 18 c[1, 9] + 18 c[1, 10] + 
 
>     55 c[2, 0] + 55 c[2, 1] + 55 c[2, 2] + 55 c[2, 3] + 55 c[2, 4] + 55 c[2, 5] + 55 c[2, 6] + 
 
>     55 c[2, 7] + 55 c[2, 8] + 55 c[2, 9] + 55 c[2, 10] + 150 c[3, 0] + 150 c[3, 1] + 150 c[3, 2] + 
 
>     150 c[3, 3] + 150 c[3, 4] + 150 c[3, 5] + 150 c[3, 6] + 150 c[3, 7] + 150 c[3, 8] + 
 
>     150 c[3, 9] + 150 c[3, 10], 18 c[0, 0] + 36 c[0, 1] + 72 c[0, 2] + 144 c[0, 3] + 288 c[0, 4] + 
 
>     576 c[0, 5] + 1152 c[0, 6] + 2304 c[0, 7] + 4608 c[0, 8] + 9216 c[0, 9] + 18432 c[0, 10] + 
 
>     55 c[1, 0] + 110 c[1, 1] + 220 c[1, 2] + 440 c[1, 3] + 880 c[1, 4] + 1760 c[1, 5] + 
 
>     3520 c[1, 6] + 7040 c[1, 7] + 14080 c[1, 8] + 28160 c[1, 9] + 56320 c[1, 10] + 150 c[2, 0] + 
 
>     300 c[2, 1] + 600 c[2, 2] + 1200 c[2, 3] + 2400 c[2, 4] + 4800 c[2, 5] + 9600 c[2, 6] + 
 
>     19200 c[2, 7] + 38400 c[2, 8] + 76800 c[2, 9] + 153600 c[2, 10] + 409 c[3, 0] + 818 c[3, 1] + 
 
>     1636 c[3, 2] + 3272 c[3, 3] + 6544 c[3, 4] + 13088 c[3, 5] + 26176 c[3, 6] + 52352 c[3, 7] + 
 
>     104704 c[3, 8] + 209408 c[3, 9] + 418816 c[3, 10], 
 
>    55 c[0, 0] + 165 c[0, 1] + 495 c[0, 2] + 1485 c[0, 3] + 4455 c[0, 4] + 13365 c[0, 5] + 
 
>     40095 c[0, 6] + 120285 c[0, 7] + 360855 c[0, 8] + 1082565 c[0, 9] + 3247695 c[0, 10] + 
 
>     150 c[1, 0] + 450 c[1, 1] + 1350 c[1, 2] + 4050 c[1, 3] + 12150 c[1, 4] + 36450 c[1, 5] + 
 
>     109350 c[1, 6] + 328050 c[1, 7] + 984150 c[1, 8] + 2952450 c[1, 9] + 8857350 c[1, 10] + 
 
>     409 c[2, 0] + 1227 c[2, 1] + 3681 c[2, 2] + 11043 c[2, 3] + 33129 c[2, 4] + 99387 c[2, 5] + 
 
>     298161 c[2, 6] + 894483 c[2, 7] + 2683449 c[2, 8] + 8050347 c[2, 9] + 24151041 c[2, 10] + 
 
>     1204 c[3, 0] + 3612 c[3, 1] + 10836 c[3, 2] + 32508 c[3, 3] + 97524 c[3, 4] + 292572 c[3, 5] + 
 
>     877716 c[3, 6] + 2633148 c[3, 7] + 7899444 c[3, 8] + 23698332 c[3, 9] + 71094996 c[3, 10], 
 
>    150 c[0, 0] + 600 c[0, 1] + 2400 c[0, 2] + 9600 c[0, 3] + 38400 c[0, 4] + 153600 c[0, 5] + 
 
>     614400 c[0, 6] + 2457600 c[0, 7] + 9830400 c[0, 8] + 39321600 c[0, 9] + 157286400 c[0, 10] + 
 
>     409 c[1, 0] + 1636 c[1, 1] + 6544 c[1, 2] + 26176 c[1, 3] + 104704 c[1, 4] + 418816 c[1, 5] + 
 
>     1675264 c[1, 6] + 6701056 c[1, 7] + 26804224 c[1, 8] + 107216896 c[1, 9] + 428867584 c[1, 10] + 
 
>     1204 c[2, 0] + 4816 c[2, 1] + 19264 c[2, 2] + 77056 c[2, 3] + 308224 c[2, 4] + 1232896 c[2, 5] + 
 
>     4931584 c[2, 6] + 19726336 c[2, 7] + 78905344 c[2, 8] + 315621376 c[2, 9] + 
 
>     1262485504 c[2, 10] + 3903 c[3, 0] + 15612 c[3, 1] + 62448 c[3, 2] + 249792 c[3, 3] + 
 
>     999168 c[3, 4] + 3996672 c[3, 5] + 15986688 c[3, 6] + 63946752 c[3, 7] + 255787008 c[3, 8] + 
 
>     1023148032 c[3, 9] + 4092592128 c[3, 10], 
 
>    409 c[0, 0] + 2045 c[0, 1] + 10225 c[0, 2] + 51125 c[0, 3] + 255625 c[0, 4] + 1278125 c[0, 5] + 
 
>     6390625 c[0, 6] + 31953125 c[0, 7] + 159765625 c[0, 8] + 798828125 c[0, 9] + 
 
>     3994140625 c[0, 10] + 1204 c[1, 0] + 6020 c[1, 1] + 30100 c[1, 2] + 150500 c[1, 3] + 
 
>     752500 c[1, 4] + 3762500 c[1, 5] + 18812500 c[1, 6] + 94062500 c[1, 7] + 470312500 c[1, 8] + 
 
>     2351562500 c[1, 9] + 11757812500 c[1, 10] + 3903 c[2, 0] + 19515 c[2, 1] + 97575 c[2, 2] + 
 
>     487875 c[2, 3] + 2439375 c[2, 4] + 12196875 c[2, 5] + 60984375 c[2, 6] + 304921875 c[2, 7] + 
 
>     1524609375 c[2, 8] + 7623046875 c[2, 9] + 38115234375 c[2, 10] + 13638 c[3, 0] + 68190 c[3, 1] + 
 
>     340950 c[3, 2] + 1704750 c[3, 3] + 8523750 c[3, 4] + 42618750 c[3, 5] + 213093750 c[3, 6] + 
 
>     1065468750 c[3, 7] + 5327343750 c[3, 8] + 26636718750 c[3, 9] + 133183593750 c[3, 10], 
 
>    1204 c[0, 0] + 7224 c[0, 1] + 43344 c[0, 2] + 260064 c[0, 3] + 1560384 c[0, 4] + 
 
>     9362304 c[0, 5] + 56173824 c[0, 6] + 337042944 c[0, 7] + 2022257664 c[0, 8] + 
 
>     12133545984 c[0, 9] + 72801275904 c[0, 10] + 3903 c[1, 0] + 23418 c[1, 1] + 140508 c[1, 2] + 
 
>     843048 c[1, 3] + 5058288 c[1, 4] + 30349728 c[1, 5] + 182098368 c[1, 6] + 1092590208 c[1, 7] + 
 
>     6555541248 c[1, 8] + 39333247488 c[1, 9] + 235999484928 c[1, 10] + 13638 c[2, 0] + 
 
>     81828 c[2, 1] + 490968 c[2, 2] + 2945808 c[2, 3] + 17674848 c[2, 4] + 106049088 c[2, 5] + 
 
>     636294528 c[2, 6] + 3817767168 c[2, 7] + 22906603008 c[2, 8] + 137439618048 c[2, 9] + 
 
>     824637708288 c[2, 10] + 49861 c[3, 0] + 299166 c[3, 1] + 1794996 c[3, 2] + 10769976 c[3, 3] + 
 
>     64619856 c[3, 4] + 387719136 c[3, 5] + 2326314816 c[3, 6] + 13957888896 c[3, 7] + 
 
>     83747333376 c[3, 8] + 502484000256 c[3, 9] + 3014904001536 c[3, 10], 
 
>    3903 c[0, 0] + 27321 c[0, 1] + 191247 c[0, 2] + 1338729 c[0, 3] + 9371103 c[0, 4] + 
 
>     65597721 c[0, 5] + 459184047 c[0, 6] + 3214288329 c[0, 7] + 22500018303 c[0, 8] + 
 
>     157500128121 c[0, 9] + 1102500896847 c[0, 10] + 13638 c[1, 0] + 95466 c[1, 1] + 668262 c[1, 2] + 
 
>     4677834 c[1, 3] + 32744838 c[1, 4] + 229213866 c[1, 5] + 1604497062 c[1, 6] + 
 
>     11231479434 c[1, 7] + 78620356038 c[1, 8] + 550342492266 c[1, 9] + 3852397445862 c[1, 10] + 
 
>     49861 c[2, 0] + 349027 c[2, 1] + 2443189 c[2, 2] + 17102323 c[2, 3] + 119716261 c[2, 4] + 
 
>     838013827 c[2, 5] + 5866096789 c[2, 6] + 41062677523 c[2, 7] + 287438742661 c[2, 8] + 
 
>     2012071198627 c[2, 9] + 14084498390389 c[2, 10] + 186780 c[3, 0] + 1307460 c[3, 1] + 
 
>     9152220 c[3, 2] + 64065540 c[3, 3] + 448458780 c[3, 4] + 3139211460 c[3, 5] + 
 
>     21974480220 c[3, 6] + 153821361540 c[3, 7] + 1076749530780 c[3, 8] + 7537246715460 c[3, 9] + 
 
>     52760727008220 c[3, 10], 13638 c[0, 0] + 109104 c[0, 1] + 872832 c[0, 2] + 6982656 c[0, 3] + 
 
>     55861248 c[0, 4] + 446889984 c[0, 5] + 3575119872 c[0, 6] + 28600958976 c[0, 7] + 
 
>     228807671808 c[0, 8] + 1830461374464 c[0, 9] + 14643690995712 c[0, 10] + 49861 c[1, 0] + 
 
>     398888 c[1, 1] + 3191104 c[1, 2] + 25528832 c[1, 3] + 204230656 c[1, 4] + 1633845248 c[1, 5] + 
 
>     13070761984 c[1, 6] + 104566095872 c[1, 7] + 836528766976 c[1, 8] + 6692230135808 c[1, 9] + 
 
>     53537841086464 c[1, 10] + 186780 c[2, 0] + 1494240 c[2, 1] + 11953920 c[2, 2] + 
 
>     95631360 c[2, 3] + 765050880 c[2, 4] + 6120407040 c[2, 5] + 48963256320 c[2, 6] + 
 
>     391706050560 c[2, 7] + 3133648404480 c[2, 8] + 25069187235840 c[2, 9] + 
 
>     200553497886720 c[2, 10] + 708811 c[3, 0] + 5670488 c[3, 1] + 45363904 c[3, 2] + 
 
>     362911232 c[3, 3] + 2903289856 c[3, 4] + 23226318848 c[3, 5] + 185810550784 c[3, 6] + 
 
>     1486484406272 c[3, 7] + 11891875250176 c[3, 8] + 95135002001408 c[3, 9] + 
 
>     761080016011264 c[3, 10], 49861 c[0, 0] + 448749 c[0, 1] + 4038741 c[0, 2] + 36348669 c[0, 3] + 
 
>     327138021 c[0, 4] + 2944242189 c[0, 5] + 26498179701 c[0, 6] + 238483617309 c[0, 7] + 
 
>     2146352555781 c[0, 8] + 19317173002029 c[0, 9] + 173854557018261 c[0, 10] + 186780 c[1, 0] + 
 
>     1681020 c[1, 1] + 15129180 c[1, 2] + 136162620 c[1, 3] + 1225463580 c[1, 4] + 
 
>     11029172220 c[1, 5] + 99262549980 c[1, 6] + 893362949820 c[1, 7] + 8040266548380 c[1, 8] + 
 
>     72362398935420 c[1, 9] + 651261590418780 c[1, 10] + 708811 c[2, 0] + 6379299 c[2, 1] + 
 
>     57413691 c[2, 2] + 516723219 c[2, 3] + 4650508971 c[2, 4] + 41854580739 c[2, 5] + 
 
>     376691226651 c[2, 6] + 3390221039859 c[2, 7] + 30511989358731 c[2, 8] + 
 
>     274607904228579 c[2, 9] + 2471471138057211 c[2, 10] + 2709980 c[3, 0] + 24389820 c[3, 1] + 
 
>     219508380 c[3, 2] + 1975575420 c[3, 3] + 17780178780 c[3, 4] + 160021609020 c[3, 5] + 
 
>     1440194481180 c[3, 6] + 12961750330620 c[3, 7] + 116655752975580 c[3, 8] + 
 
>     1049901776780220 c[3, 9] + 9449115991021980 c[3, 10], 
 
>    186780 c[0, 0] + 1867800 c[0, 1] + 18678000 c[0, 2] + 186780000 c[0, 3] + 1867800000 c[0, 4] + 
 
>     18678000000 c[0, 5] + 186780000000 c[0, 6] + 1867800000000 c[0, 7] + 18678000000000 c[0, 8] + 
 
>     186780000000000 c[0, 9] + 1867800000000000 c[0, 10] + 708811 c[1, 0] + 7088110 c[1, 1] + 
 
>     70881100 c[1, 2] + 708811000 c[1, 3] + 7088110000 c[1, 4] + 70881100000 c[1, 5] + 
 
>     708811000000 c[1, 6] + 7088110000000 c[1, 7] + 70881100000000 c[1, 8] + 
 
>     708811000000000 c[1, 9] + 7088110000000000 c[1, 10] + 2709980 c[2, 0] + 27099800 c[2, 1] + 
 
>     270998000 c[2, 2] + 2709980000 c[2, 3] + 27099800000 c[2, 4] + 270998000000 c[2, 5] + 
 
>     2709980000000 c[2, 6] + 27099800000000 c[2, 7] + 270998000000000 c[2, 8] + 
 
>     2709980000000000 c[2, 9] + 27099800000000000 c[2, 10] + 10410989 c[3, 0] + 104109890 c[3, 1] + 
 
>     1041098900 c[3, 2] + 10410989000 c[3, 3] + 104109890000 c[3, 4] + 1041098900000 c[3, 5] + 
 
>     10410989000000 c[3, 6] + 104109890000000 c[3, 7] + 1041098900000000 c[3, 8] + 
 
>     10410989000000000 c[3, 9] + 104109890000000000 c[3, 10], 
 
>    708811 c[0, 0] + 7796921 c[0, 1] + 85766131 c[0, 2] + 943427441 c[0, 3] + 10377701851 c[0, 4] + 
 
>     114154720361 c[0, 5] + 1255701923971 c[0, 6] + 13812721163681 c[0, 7] + 
 
>     151939932800491 c[0, 8] + 1671339260805401 c[0, 9] + 18384731868859411 c[0, 10] + 
 
>     2709980 c[1, 0] + 29809780 c[1, 1] + 327907580 c[1, 2] + 3606983380 c[1, 3] + 
 
>     39676817180 c[1, 4] + 436444988980 c[1, 5] + 4800894878780 c[1, 6] + 52809843666580 c[1, 7] + 
 
>     580908280332380 c[1, 8] + 6389991083656180 c[1, 9] + 70289901920217980 c[1, 10] + 
 
>     10410989 c[2, 0] + 114520879 c[2, 1] + 1259729669 c[2, 2] + 13857026359 c[2, 3] + 
 
>     152427289949 c[2, 4] + 1676700189439 c[2, 5] + 18443702083829 c[2, 6] + 
 
>     202880722922119 c[2, 7] + 2231687952143309 c[2, 8] + 24548567473576399 c[2, 9] + 
 
>     270034242209340389 c[2, 10] + 40135728 c[3, 0] + 441493008 c[3, 1] + 4856423088 c[3, 2] + 
 
>     53420653968 c[3, 3] + 587627193648 c[3, 4] + 6463899130128 c[3, 5] + 71102890431408 c[3, 6] + 
 
>     782131794745488 c[3, 7] + 8603449742200368 c[3, 8] + 94637947164204048 c[3, 9] + 
 
>     1041017418806244528 c[3, 10], 2709980 c[0, 0] + 32519760 c[0, 1] + 390237120 c[0, 2] + 
 
>     4682845440 c[0, 3] + 56194145280 c[0, 4] + 674329743360 c[0, 5] + 8091956920320 c[0, 6] + 
 
>     97103483043840 c[0, 7] + 1165241796526080 c[0, 8] + 13982901558312960 c[0, 9] + 
 
>     167794818699755520 c[0, 10] + 10410989 c[1, 0] + 124931868 c[1, 1] + 1499182416 c[1, 2] + 
 
>     17990188992 c[1, 3] + 215882267904 c[1, 4] + 2590587214848 c[1, 5] + 31087046578176 c[1, 6] + 
 
>     373044558938112 c[1, 7] + 4476534707257344 c[1, 8] + 53718416487088128 c[1, 9] + 
 
>     644620997845057536 c[1, 10] + 40135728 c[2, 0] + 481628736 c[2, 1] + 5779544832 c[2, 2] + 
 
>     69354537984 c[2, 3] + 832254455808 c[2, 4] + 9987053469696 c[2, 5] + 119844641636352 c[2, 6] + 
 
>     1438135699636224 c[2, 7] + 17257628395634688 c[2, 8] + 207091540747616256 c[2, 9] + 
 
>     2485098488971395072 c[2, 10] + 155153663 c[3, 0] + 1861843956 c[3, 1] + 22342127472 c[3, 2] + 
 
>     268105529664 c[3, 3] + 3217266355968 c[3, 4] + 38607196271616 c[3, 5] + 
 
>     463286355259392 c[3, 6] + 5559436263112704 c[3, 7] + 66713235157352448 c[3, 8] + 
 
>     800558821888229376 c[3, 9] + 9606705862658752512 c[3, 10], 
 
>    10410989 c[0, 0] + 135342857 c[0, 1] + 1759457141 c[0, 2] + 22872942833 c[0, 3] + 
 
>     297348256829 c[0, 4] + 3865527338777 c[0, 5] + 50251855404101 c[0, 6] + 
 
>     653274120253313 c[0, 7] + 8492563563293069 c[0, 8] + 110403326322809897 c[0, 9] + 
 
>     1435243242196528661 c[0, 10] + 40135728 c[1, 0] + 521764464 c[1, 1] + 6782938032 c[1, 2] + 
 
>     88178194416 c[1, 3] + 1146316527408 c[1, 4] + 14902114856304 c[1, 5] + 193727493131952 c[1, 6] + 
 
>     2518457410715376 c[1, 7] + 32739946339299888 c[1, 8] + 425619302410898544 c[1, 9] + 
 
>     5533050931341681072 c[1, 10] + 155153663 c[2, 0] + 2016997619 c[2, 1] + 26220969047 c[2, 2] + 
 
>     340872597611 c[2, 3] + 4431343768943 c[2, 4] + 57607468996259 c[2, 5] + 
 
>     748897096951367 c[2, 6] + 9735662260367771 c[2, 7] + 126563609384781023 c[2, 8] + 
 
>     1645326922002153299 c[2, 9] + 21389249986027992887 c[2, 10] + 601150022 c[3, 0] + 
 
>     7814950286 c[3, 1] + 101594353718 c[3, 2] + 1320726598334 c[3, 3] + 17169445778342 c[3, 4] + 
 
>     223202795118446 c[3, 5] + 2901636336539798 c[3, 6] + 37721272375017374 c[3, 7] + 
 
>     490376540875225862 c[3, 8] + 6374895031377936206 c[3, 9] + 82873635407913170678 c[3, 10], 
 
>    40135728 c[0, 0] + 561900192 c[0, 1] + 7866602688 c[0, 2] + 110132437632 c[0, 3] + 
 
>     1541854126848 c[0, 4] + 21585957775872 c[0, 5] + 302203408862208 c[0, 6] + 
 
>     4230847724070912 c[0, 7] + 59231868136992768 c[0, 8] + 829246153917898752 c[0, 9] + 
 
>     11609446154850582528 c[0, 10] + 155153663 c[1, 0] + 2172151282 c[1, 1] + 30410117948 c[1, 2] + 
 
>     425741651272 c[1, 3] + 5960383117808 c[1, 4] + 83445363649312 c[1, 5] + 
 
>     1168235091090368 c[1, 6] + 16355291275265152 c[1, 7] + 228974077853712128 c[1, 8] + 
 
>     3205637089951969792 c[1, 9] + 44878919259327577088 c[1, 10] + 601150022 c[2, 0] + 
 
>     8416100308 c[2, 1] + 117825404312 c[2, 2] + 1649555660368 c[2, 3] + 23093779245152 c[2, 4] + 
 
>     323312909432128 c[2, 5] + 4526380732049792 c[2, 6] + 63369330248697088 c[2, 7] + 
 
>     887170623481759232 c[2, 8] + 12420388728744629248 c[2, 9] + 173885442202424809472 c[2, 10] + 
 
>     2333742205 c[3, 0] + 32672390870 c[3, 1] + 457413472180 c[3, 2] + 6403788610520 c[3, 3] + 
 
>     89653040547280 c[3, 4] + 1255142567661920 c[3, 5] + 17571995947266880 c[3, 6] + 
 
>     246007943261736320 c[3, 7] + 3444111205664308480 c[3, 8] + 48217556879300318720 c[3, 9] + 
 
>     675045796310204462080 c[3, 10], 155153663 c[0, 0] + 2327304945 c[0, 1] + 34909574175 c[0, 2] + 
 
>     523643612625 c[0, 3] + 7854654189375 c[0, 4] + 117819812840625 c[0, 5] + 
 
>     1767297192609375 c[0, 6] + 26509457889140625 c[0, 7] + 397641868337109375 c[0, 8] + 
 
>     5964628025056640625 c[0, 9] + 89469420375849609375 c[0, 10] + 601150022 c[1, 0] + 
 
>     9017250330 c[1, 1] + 135258754950 c[1, 2] + 2028881324250 c[1, 3] + 30433219863750 c[1, 4] + 
 
>     456498297956250 c[1, 5] + 6847474469343750 c[1, 6] + 102712117040156250 c[1, 7] + 
 
>     1540681755602343750 c[1, 8] + 23110226334035156250 c[1, 9] + 346653395010527343750 c[1, 10] + 
 
>     2333742205 c[2, 0] + 35006133075 c[2, 1] + 525091996125 c[2, 2] + 7876379941875 c[2, 3] + 
 
>     118145699128125 c[2, 4] + 1772185486921875 c[2, 5] + 26582782303828125 c[2, 6] + 
 
>     398741734557421875 c[2, 7] + 5981126018361328125 c[2, 8] + 89716890275419921875 c[2, 9] + 
 
>     1345753354131298828125 c[2, 10] + 9075403276 c[3, 0] + 136131049140 c[3, 1] + 
 
>     2041965737100 c[3, 2] + 30629486056500 c[3, 3] + 459442290847500 c[3, 4] + 
 
>     6891634362712500 c[3, 5] + 103374515440687500 c[3, 6] + 1550617731610312500 c[3, 7] + 
 
>     23259265974154687500 c[3, 8] + 348888989612320312500 c[3, 9] + 5233334844184804687500 c[3, 10], 
 
>    601150022 c[0, 0] + 9618400352 c[0, 1] + 153894405632 c[0, 2] + 2462310490112 c[0, 3] + 
 
>     39396967841792 c[0, 4] + 630351485468672 c[0, 5] + 10085623767498752 c[0, 6] + 
 
>     161369980279980032 c[0, 7] + 2581919684479680512 c[0, 8] + 41310714951674888192 c[0, 9] + 
 
>     660971439226798211072 c[0, 10] + 2333742205 c[1, 0] + 37339875280 c[1, 1] + 
 
>     597438004480 c[1, 2] + 9559008071680 c[1, 3] + 152944129146880 c[1, 4] + 
 
>     2447106066350080 c[1, 5] + 39153697061601280 c[1, 6] + 626459152985620480 c[1, 7] + 
 
>     10023346447769927680 c[1, 8] + 160373543164318842880 c[1, 9] + 2565976690629101486080 c[1, 10] + 
 
>     9075403276 c[2, 0] + 145206452416 c[2, 1] + 2323303238656 c[2, 2] + 37172851818496 c[2, 3] + 
 
>     594765629095936 c[2, 4] + 9516250065534976 c[2, 5] + 152260001048559616 c[2, 6] + 
 
>     2436160016776953856 c[2, 7] + 38978560268431261696 c[2, 8] + 623656964294900187136 c[2, 9] + 
 
>     9978511428718402994176 c[2, 10] + 35345794947 c[3, 0] + 565532719152 c[3, 1] + 
 
>     9048523506432 c[3, 2] + 144776376102912 c[3, 3] + 2316422017646592 c[3, 4] + 
 
>     37062752282345472 c[3, 5] + 593004036517527552 c[3, 6] + 9488064584280440832 c[3, 7] + 
 
>     151809033348487053312 c[3, 8] + 2428944533575792852992 c[3, 9] + 38863112537212685647872 c[3, 10]
 
>     , 2333742205 c[0, 0] + 39673617485 c[0, 1] + 674451497245 c[0, 2] + 11465675453165 c[0, 3] + 
 
>     194916482703805 c[0, 4] + 3313580205964685 c[0, 5] + 56330863501399645 c[0, 6] + 
 
>     957624679523793965 c[0, 7] + 16279619551904497405 c[0, 8] + 276753532382376455885 c[0, 9] + 
 
>     4704810050500399750045 c[0, 10] + 9075403276 c[1, 0] + 154281855692 c[1, 1] + 
 
>     2622791546764 c[1, 2] + 44587456294988 c[1, 3] + 757986757014796 c[1, 4] + 
 
>     12885774869251532 c[1, 5] + 219058172777276044 c[1, 6] + 3723988937213692748 c[1, 7] + 
 
>     63307811932632776716 c[1, 8] + 1076232802854757204172 c[1, 9] + 
 
>     18295957648530872470924 c[1, 10] + 35345794947 c[2, 0] + 600878514099 c[2, 1] + 
 
>     10214934739683 c[2, 2] + 173653890574611 c[2, 3] + 2952116139768387 c[2, 4] + 
 
>     50185974376062579 c[2, 5] + 853161564393063843 c[2, 6] + 14503746594682085331 c[2, 7] + 
 
>     246563692109595450627 c[2, 8] + 4191582765863122660659 c[2, 9] + 
 
>     71256907019673085231203 c[2, 10] + 137847585396 c[3, 0] + 2343408951732 c[3, 1] + 
 
>     39837952179444 c[3, 2] + 677245187050548 c[3, 3] + 11513168179859316 c[3, 4] + 
 
>     195723859057608372 c[3, 5] + 3327305603979342324 c[3, 6] + 56564195267648819508 c[3, 7] + 
 
>     961591319550029931636 c[3, 8] + 16347052432350508837812 c[3, 9] + 
 
>     277899891349958650242804 c[3, 10], 9075403276 c[0, 0] + 163357258968 c[0, 1] + 
 
>     2940430661424 c[0, 2] + 52927751905632 c[0, 3] + 952699534301376 c[0, 4] + 
 
>     17148591617424768 c[0, 5] + 308674649113645824 c[0, 6] + 5556143684045624832 c[0, 7] + 
 
>     100010586312821246976 c[0, 8] + 1800190553630782445568 c[0, 9] + 
 
>     32403429965354084020224 c[0, 10] + 35345794947 c[1, 0] + 636224309046 c[1, 1] + 
 
>     11452037562828 c[1, 2] + 206136676130904 c[1, 3] + 3710460170356272 c[1, 4] + 
 
>     66788283066412896 c[1, 5] + 1202189095195432128 c[1, 6] + 21639403713517778304 c[1, 7] + 
 
>     389509266843320009472 c[1, 8] + 7011166803179760170496 c[1, 9] + 
 
>     126201002457235683068928 c[1, 10] + 137847585396 c[2, 0] + 2481256537128 c[2, 1] + 
 
>     44662617668304 c[2, 2] + 803927118029472 c[2, 3] + 14470688124530496 c[2, 4] + 
 
>     260472386241548928 c[2, 5] + 4688502952347880704 c[2, 6] + 84393053142261852672 c[2, 7] + 
 
>     1519074956560713348096 c[2, 8] + 27343349218092840265728 c[2, 9] + 
 
>     492180285925671124783104 c[2, 10] + 538259980853 c[3, 0] + 9688679655354 c[3, 1] + 
 
>     174396233796372 c[3, 2] + 3139132208334696 c[3, 3] + 56504379750024528 c[3, 4] + 
 
>     1017078835500441504 c[3, 5] + 18307419039007947072 c[3, 6] + 329533542702143047296 c[3, 7] + 
 
>     5931603768638574851328 c[3, 8] + 106768867835494347323904 c[3, 9] + 
 
>     1921839621038898251830272 c[3, 10], 
 
>    35345794947 c[0, 0] + 671570103993 c[0, 1] + 12759831975867 c[0, 2] + 242436807541473 c[0, 3] + 
 
>     4606299343287987 c[0, 4] + 87519687522471753 c[0, 5] + 1662874062926963307 c[0, 6] + 
 
>     31594607195612302833 c[0, 7] + 600297536716633753827 c[0, 8] + 11405653197616041322713 c[0, 9] + 
 
>     216707410754704785131547 c[0, 10] + 137847585396 c[1, 0] + 2619104122524 c[1, 1] + 
 
>     49762978327956 c[1, 2] + 945496588231164 c[1, 3] + 17964435176392116 c[1, 4] + 
 
>     341324268351450204 c[1, 5] + 6485161098677553876 c[1, 6] + 123218060874873523644 c[1, 7] + 
 
>     2341143156622596949236 c[1, 8] + 44481719975829342035484 c[1, 9] + 
 
>     845152679540757498674196 c[1, 10] + 538259980853 c[2, 0] + 10226939636207 c[2, 1] + 
 
>     194311853087933 c[2, 2] + 3691925208670727 c[2, 3] + 70146578964743813 c[2, 4] + 
 
>     1332785000330132447 c[2, 5] + 25322915006272516493 c[2, 6] + 481135385119177813367 c[2, 7] + 
 
>     9141572317264378453973 c[2, 8] + 173689874028023190625487 c[2, 9] + 
 
>     3300107606532440621884253 c[2, 10] + 2104103168672 c[3, 0] + 39977960204768 c[3, 1] + 
 
>     759581243890592 c[3, 2] + 14432043633921248 c[3, 3] + 274208829044503712 c[3, 4] + 
 
>     5209967751845570528 c[3, 5] + 98989387285065840032 c[3, 6] + 1880798358416250960608 c[3, 7] + 
 
>     35735168809908768251552 c[3, 8] + 678968207388266596779488 c[3, 9] + 
 
>     12900395940377065338810272 c[3, 10], 
 
>    137847585396 c[0, 0] + 2756951707920 c[0, 1] + 55139034158400 c[0, 2] + 
 
>     1102780683168000 c[0, 3] + 22055613663360000 c[0, 4] + 441112273267200000 c[0, 5] + 
 
>     8822245465344000000 c[0, 6] + 176444909306880000000 c[0, 7] + 3528898186137600000000 c[0, 8] + 
 
>     70577963722752000000000 c[0, 9] + 1411559274455040000000000 c[0, 10] + 538259980853 c[1, 0] + 
 
>     10765199617060 c[1, 1] + 215303992341200 c[1, 2] + 4306079846824000 c[1, 3] + 
 
>     86121596936480000 c[1, 4] + 1722431938729600000 c[1, 5] + 34448638774592000000 c[1, 6] + 
 
>     688972775491840000000 c[1, 7] + 13779455509836800000000 c[1, 8] + 
 
>     275589110196736000000000 c[1, 9] + 5511782203934720000000000 c[1, 10] + 2104103168672 c[2, 0] + 
 
>     42082063373440 c[2, 1] + 841641267468800 c[2, 2] + 16832825349376000 c[2, 3] + 
 
>     336656506987520000 c[2, 4] + 6733130139750400000 c[2, 5] + 134662602795008000000 c[2, 6] + 
 
>     2693252055900160000000 c[2, 7] + 53865041118003200000000 c[2, 8] + 
 
>     1077300822360064000000000 c[2, 9] + 21546016447201280000000000 c[2, 10] + 
 
>     8233439128375 c[3, 0] + 164668782567500 c[3, 1] + 3293375651350000 c[3, 2] + 
 
>     65867513027000000 c[3, 3] + 1317350260540000000 c[3, 4] + 26347005210800000000 c[3, 5] + 
 
>     526940104216000000000 c[3, 6] + 10538802084320000000000 c[3, 7] + 
 
>     210776041686400000000000 c[3, 8] + 4215520833728000000000000 c[3, 9] + 
 
>     84310416674560000000000000 c[3, 10], 
 
>    538259980853 c[0, 0] + 11303459597913 c[0, 1] + 237372651556173 c[0, 2] + 
 
>     4984825682679633 c[0, 3] + 104681339336272293 c[0, 4] + 2198308126061718153 c[0, 5] + 
 
>     46164470647296081213 c[0, 6] + 969453883593217705473 c[0, 7] + 20358531555457571814933 c[0, 8] + 
 
>     427529162664609008113593 c[0, 9] + 8978112415956789170385453 c[0, 10] + 2104103168672 c[1, 0] + 
 
>     44186166542112 c[1, 1] + 927909497384352 c[1, 2] + 19486099445071392 c[1, 3] + 
 
>     409208088346499232 c[1, 4] + 8593369855276483872 c[1, 5] + 180460766960806161312 c[1, 6] + 
 
>     3789676106176929387552 c[1, 7] + 79583198229715517138592 c[1, 8] + 
 
>     1671247162824025859910432 c[1, 9] + 35096190419304543058119072 c[1, 10] + 
 
>     8233439128375 c[2, 0] + 172902221695875 c[2, 1] + 3630946655613375 c[2, 2] + 
 
>     76249879767880875 c[2, 3] + 1601247475125498375 c[2, 4] + 33626196977635465875 c[2, 5] + 
 
>     706150136530344783375 c[2, 6] + 14829152867137240450875 c[2, 7] + 
 
>     311412210209882049468375 c[2, 8] + 6539656414407523038835875 c[2, 9] + 
 
>     137332784702557983815553375 c[2, 10] + 32247620474140 c[3, 0] + 677200029956940 c[3, 1] + 
 
>     14221200629095740 c[3, 2] + 298645213211010540 c[3, 3] + 6271549477431221340 c[3, 4] + 
 
>     131702539026055648140 c[3, 5] + 2765753319547168610940 c[3, 6] + 
 
>     58080819710490540829740 c[3, 7] + 1219697213920301357424540 c[3, 8] + 
 
>     25613641492326328505915340 c[3, 9] + 537886471338852898624222140 c[3, 10], 
 
>    2104103168672 c[0, 0] + 46290269710784 c[0, 1] + 1018385933637248 c[0, 2] + 
 
>     22404490540019456 c[0, 3] + 492898791880428032 c[0, 4] + 10843773421369416704 c[0, 5] + 
 
>     238563015270127167488 c[0, 6] + 5248386335942797684736 c[0, 7] + 
 
>     115464499390741549064192 c[0, 8] + 2540218986596314079412224 c[0, 9] + 
 
>     55884817705118909747068928 c[0, 10] + 8233439128375 c[1, 0] + 181135660824250 c[1, 1] + 
 
>     3984984538133500 c[1, 2] + 87669659838937000 c[1, 3] + 1928732516456614000 c[1, 4] + 
 
>     42432115362045508000 c[1, 5] + 933506537965001176000 c[1, 6] + 20537143835230025872000 c[1, 7] + 
 
>     451817164375060569184000 c[1, 8] + 9939977616251332522048000 c[1, 9] + 
 
>     218679507557529315485056000 c[1, 10] + 32247620474140 c[2, 0] + 709447650431080 c[2, 1] + 
 
>     15607848309483760 c[2, 2] + 343372662808642720 c[2, 3] + 7554198581790139840 c[2, 4] + 
 
>     166192368799383076480 c[2, 5] + 3656232113586427682560 c[2, 6] + 
 
>     80437106498901409016320 c[2, 7] + 1769616342975830998359040 c[2, 8] + 
 
>     38931559545468281963898880 c[2, 9] + 856494310000302203205775360 c[2, 10] + 
 
>     126410640007809 c[3, 0] + 2781034080171798 c[3, 1] + 61182749763779556 c[3, 2] + 
 
>     1346020494803150232 c[3, 3] + 29612450885669305104 c[3, 4] + 651473919484724712288 c[3, 5] + 
 
>     14332426228663943670336 c[3, 6] + 315313377030606760747392 c[3, 7] + 
 
>     6936894294673348736442624 c[3, 8] + 152611674482813672201737728 c[3, 9] + 
 
>     3357456838621900788438230016 c[3, 10], 
 
>    8233439128375 c[0, 0] + 189369099952625 c[0, 1] + 4355489298910375 c[0, 2] + 
 
>     100176253874938625 c[0, 3] + 2304053839123588375 c[0, 4] + 52993238299842532625 c[0, 5] + 
 
>     1218844480896378250375 c[0, 6] + 28033423060616699758625 c[0, 7] + 
 
>     644768730394184094448375 c[0, 8] + 14829680799066234172312625 c[0, 9] + 
 
>     341082658378523385963190375 c[0, 10] + 32247620474140 c[1, 0] + 741695270905220 c[1, 1] + 
 
>     17058991230820060 c[1, 2] + 392356798308861380 c[1, 3] + 9024206361103811740 c[1, 4] + 
 
>     207556746305387670020 c[1, 5] + 4773805165023916410460 c[1, 6] + 
 
>     109797518795550077440580 c[1, 7] + 2525342932297651781133340 c[1, 8] + 
 
>     58082887442845990966066820 c[1, 9] + 1335906411185457792219536860 c[1, 10] + 
 
>     126410640007809 c[2, 0] + 2907444720179607 c[2, 1] + 66871228564130961 c[2, 2] + 
 
>     1538038256975012103 c[2, 3] + 35374879910425278369 c[2, 4] + 813622237939781402487 c[2, 5] + 
 
>     18713311472614972257201 c[2, 6] + 430406163870144361915623 c[2, 7] + 
 
>     9899341769013320324059329 c[2, 8] + 227684860687306367453364567 c[2, 9] + 
 
>     5236751795808046451427385041 c[2, 10] + 495918600074544 c[3, 0] + 11406127801714512 c[3, 1] + 
 
>     262340939439433776 c[3, 2] + 6033841607106976848 c[3, 3] + 138778356963460467504 c[3, 4] + 
 
>     3191902210159590752592 c[3, 5] + 73413750833670587309616 c[3, 6] + 
 
>     1688516269174423508121168 c[3, 7] + 38835874191011740686786864 c[3, 8] + 
 
>     893225106393270035796097872 c[3, 9] + 20544177447045210823310251056 c[3, 10], 
 
>    32247620474140 c[0, 0] + 773942891379360 c[0, 1] + 18574629393104640 c[0, 2] + 
 
>     445791105434511360 c[0, 3] + 10698986530428272640 c[0, 4] + 256775676730278543360 c[0, 5] + 
 
>     6162616241526685040640 c[0, 6] + 147902789796640440975360 c[0, 7] + 
 
>     3549666955119370583408640 c[0, 8] + 85192006922864894001807360 c[0, 9] + 
 
>     2044608166148757456043376640 c[0, 10] + 126410640007809 c[1, 0] + 3033855360187416 c[1, 1] + 
 
>     72812528644497984 c[1, 2] + 1747500687467951616 c[1, 3] + 41940016499230838784 c[1, 4] + 
 
>     1006560395981540130816 c[1, 5] + 24157449503556963139584 c[1, 6] + 
 
>     579778788085367115350016 c[1, 7] + 13914690914048810768400384 c[1, 8] + 
 
>     333952581937171458441609216 c[1, 9] + 8014861966492115002598621184 c[1, 10] + 
 
>     495918600074544 c[2, 0] + 11902046401789056 c[2, 1] + 285649113642937344 c[2, 2] + 
 
>     6855578727430496256 c[2, 3] + 164533889458331910144 c[2, 4] + 3948813346999965843456 c[2, 5] + 
 
>     94771520327999180242944 c[2, 6] + 2274516487871980325830656 c[2, 7] + 
 
>     54588395708927527819935744 c[2, 8] + 1310121497014260667678457856 c[2, 9] + 
 
>     31442915928342256024282988544 c[2, 10] + 1946939559885523 c[3, 0] + 46726549437252552 c[3, 1] + 
 
>     1121437186494061248 c[3, 2] + 26914492475857469952 c[3, 3] + 645947819420579278848 c[3, 4] + 
 
>     15502747666093902692352 c[3, 5] + 372065943986253664616448 c[3, 6] + 
 
>     8929582655670087950794752 c[3, 7] + 214309983736082110819074048 c[3, 8] + 
 
>     5143439609665970659657777152 c[3, 9] + 123442550631983295831786651648 c[3, 10], 
 
>    126410640007809 c[0, 0] + 3160266000195225 c[0, 1] + 79006650004880625 c[0, 2] + 
 
>     1975166250122015625 c[0, 3] + 49379156253050390625 c[0, 4] + 1234478906326259765625 c[0, 5] + 
 
>     30861972658156494140625 c[0, 6] + 771549316453912353515625 c[0, 7] + 
 
>     19288732911347808837890625 c[0, 8] + 482218322783695220947265625 c[0, 9] + 
 
>     12055458069592380523681640625 c[0, 10] + 495918600074544 c[1, 0] + 12397965001863600 c[1, 1] + 
 
>     309949125046590000 c[1, 2] + 7748728126164750000 c[1, 3] + 193718203154118750000 c[1, 4] + 
 
>     4842955078852968750000 c[1, 5] + 121073876971324218750000 c[1, 6] + 
 
>     3026846924283105468750000 c[1, 7] + 75671173107077636718750000 c[1, 8] + 
 
>     1891779327676940917968750000 c[1, 9] + 47294483191923522949218750000 c[1, 10] + 
 
>     1946939559885523 c[2, 0] + 48673488997138075 c[2, 1] + 1216837224928451875 c[2, 2] + 
 
>     30420930623211296875 c[2, 3] + 760523265580282421875 c[2, 4] + 19013081639507060546875 c[2, 5] + 
 
>     475327040987676513671875 c[2, 6] + 11883176024691912841796875 c[2, 7] + 
 
>     297079400617297821044921875 c[2, 8] + 7426985015432445526123046875 c[2, 9] + 
 
>     185674625385811138153076171875 c[2, 10] + 7648690869217848 c[3, 0] + 
 
>     191217271730446200 c[3, 1] + 4780431793261155000 c[3, 2] + 119510794831528875000 c[3, 3] + 
 
>     2987769870788221875000 c[3, 4] + 74694246769705546875000 c[3, 5] + 
 
>     1867356169242638671875000 c[3, 6] + 46683904231065966796875000 c[3, 7] + 
 
>     1167097605776649169921875000 c[3, 8] + 29177440144416229248046875000 c[3, 9] + 
 
>     729436003610405731201171875000 c[3, 10], 
 
>    495918600074544 c[0, 0] + 12893883601938144 c[0, 1] + 335240973650391744 c[0, 2] + 
 
>     8716265314910185344 c[0, 3] + 226622898187664818944 c[0, 4] + 5892195352879285292544 c[0, 5] + 
 
>     153197079174861417606144 c[0, 6] + 3983124058546396857759744 c[0, 7] + 
 
>     103561225522206318301753344 c[0, 8] + 2692591863577364275845586944 c[0, 9] + 
 
>     70007388453011471171985260544 c[0, 10] + 1946939559885523 c[1, 0] + 50620428557023598 c[1, 1] + 
 
>     1316131142482613548 c[1, 2] + 34219409704547952248 c[1, 3] + 889704652318246758448 c[1, 4] + 
 
>     23132320960274415719648 c[1, 5] + 601440344967134808710848 c[1, 6] + 
 
>     15637448969145505026482048 c[1, 7] + 406573673197783130688533248 c[1, 8] + 
 
>     10570915503142361397901864448 c[1, 9] + 274843803081701396345448475648 c[1, 10] + 
 
>     7648690869217848 c[2, 0] + 198865962599664048 c[2, 1] + 5170515027591265248 c[2, 2] + 
 
>     134433390717372896448 c[2, 3] + 3495268158651695307648 c[2, 4] + 
 
>     90876972124944077998848 c[2, 5] + 2362801275248546027970048 c[2, 6] + 
 
>     61432833156462196727221248 c[2, 7] + 1597253662068017114907752448 c[2, 8] + 
 
>     41528595213768444987601563648 c[2, 9] + 1079743475557979569677640654848 c[2, 10] + 
 
>     30067267036436341 c[3, 0] + 781748942947344866 c[3, 1] + 20325472516630966516 c[3, 2] + 
 
>     528462285432405129416 c[3, 3] + 13740019421242533364816 c[3, 4] + 
 
>     357240504952305867485216 c[3, 5] + 9288253128759952554615616 c[3, 6] + 
 
>     241494581347758766420006016 c[3, 7] + 6278859115041727926920156416 c[3, 8] + 
 
>     163250336991084926099924066816 c[3, 9] + 4244508761768208078598025737216 c[3, 10], 
 
>    1946939559885523 c[0, 0] + 52567368116909121 c[0, 1] + 1419318939156546267 c[0, 2] + 
 
>     38321611357226749209 c[0, 3] + 1034683506645122228643 c[0, 4] + 
 
>     27936454679418300173361 c[0, 5] + 754284276344294104680747 c[0, 6] + 
 
>     20365675461295940826380169 c[0, 7] + 549873237454990402312264563 c[0, 8] + 
 
>     14846577411284740862431143201 c[0, 9] + 400857590104688003285640866427 c[0, 10] + 
 
>     7648690869217848 c[1, 0] + 206514653468881896 c[1, 1] + 5575895643659811192 c[1, 2] + 
 
>     150549182378814902184 c[1, 3] + 4064827924228002358968 c[1, 4] + 
 
>     109750353954156063692136 c[1, 5] + 2963259556762213719687672 c[1, 6] + 
 
>     80008008032579770431567144 c[1, 7] + 2160216216879653801652312888 c[1, 8] + 
 
>     58325837855750652644612447976 c[1, 9] + 1574797622105267621404536095352 c[1, 10] + 
 
>     30067267036436341 c[2, 0] + 811816209983781207 c[2, 1] + 21919037669562092589 c[2, 2] + 
 
>     591814017078176499903 c[2, 3] + 15978978461110765497381 c[2, 4] + 
 
>     431432418449990668429287 c[2, 5] + 11648675298149748047590749 c[2, 6] + 
 
>     314514233050043197284950223 c[2, 7] + 8491884292351166326693656021 c[2, 8] + 
 
>     229280875893481490820728712567 c[2, 9] + 6190583649124000252159675239309 c[2, 10] + 
 
>     118264582638630248 c[3, 0] + 3193143731243016696 c[3, 1] + 86214880743561450792 c[3, 2] + 
 
>     2327801780076159171384 c[3, 3] + 62850648062056297627368 c[3, 4] + 
 
>     1696967497675520035938936 c[3, 5] + 45818122437239040970351272 c[3, 6] + 
 
>     1237089305805454106199484344 c[3, 7] + 33401411256747260867386077288 c[3, 8] + 
 
>     901838103932176043419424086776 c[3, 9] + 24349628806168753172324450342952 c[3, 10], 
 
>    7648690869217848 c[0, 0] + 214163344338099744 c[0, 1] + 5996573641466792832 c[0, 2] + 
 
>     167904061961070199296 c[0, 3] + 4701313734909965580288 c[0, 4] + 
 
>     131636784577479036248064 c[0, 5] + 3685829968169413014945792 c[0, 6] + 
 
>     103203239108743564418482176 c[0, 7] + 2889690695044819803717500928 c[0, 8] + 
 
>     80911339461254954504090025984 c[0, 9] + 2265517504915138726114520727552 c[0, 10] + 
 
>     30067267036436341 c[1, 0] + 841883477020217548 c[1, 1] + 23572737356566091344 c[1, 2] + 
 
>     660036645983850557632 c[1, 3] + 18481026087547815613696 c[1, 4] + 
 
>     517468730451338837183488 c[1, 5] + 14489124452637487441137664 c[1, 6] + 
 
>     405695484673849648351854592 c[1, 7] + 11359473570867790153851928576 c[1, 8] + 
 
>     318065259984298124307854000128 c[1, 9] + 8905827279560347480619912003584 c[1, 10] + 
 
>     118264582638630248 c[2, 0] + 3311408313881646944 c[2, 1] + 92719432788686114432 c[2, 2] + 
 
>     2596144118083211204096 c[2, 3] + 72692035306329913714688 c[2, 4] + 
 
>     2035376988577237584011264 c[2, 5] + 56990555680162652352315392 c[2, 6] + 
 
>     1595735559044554265864830976 c[2, 7] + 44680595653247519444215267328 c[2, 8] + 
 
>     1251056678290930544438027485184 c[2, 9] + 35029586992146055244264769585152 c[2, 10] + 
 
>     465428355402774527 c[3, 0] + 13031993951277686756 c[3, 1] + 364895830635775229168 c[3, 2] + 
 
>     10217083257801706416704 c[3, 3] + 286078331218447779667712 c[3, 4] + 
 
>     8010193274116537830695936 c[3, 5] + 224285411675263059259486208 c[3, 6] + 
 
>     6279991526907365659265613824 c[3, 7] + 175839762753406238459437187072 c[3, 8] + 
 
>     4923513357095374676864241238016 c[3, 9] + 137858373998670490952198754664448 c[3, 10], 
 
>    30067267036436341 c[0, 0] + 871950744056653889 c[0, 1] + 25286571577642962781 c[0, 2] + 
 
>     733310575751645920649 c[0, 3] + 21266006696797731698821 c[0, 4] + 
 
>     616714194207134219265809 c[0, 5] + 17884711632006892358708461 c[0, 6] + 
 
>     518656637328199878402545369 c[0, 7] + 15041042482517796473673815701 c[0, 8] + 
 
>     436190231993016097736540655329 c[0, 9] + 12649516727797466834359679004541 c[0, 10] + 
 
>     118264582638630248 c[1, 0] + 3429672896520277192 c[1, 1] + 99460513999088038568 c[1, 2] + 
 
>     2884354905973553118472 c[1, 3] + 83646292273233040435688 c[1, 4] + 
 
>     2425742475923758172634952 c[1, 5] + 70346531801788987006413608 c[1, 6] + 
 
>     2040049422251880623185994632 c[1, 7] + 59161433245304538072393844328 c[1, 8] + 
 
>     1715681564113831604099421485512 c[1, 9] + 49754765359301116518883223079848 c[1, 10] + 
 
>     465428355402774527 c[2, 0] + 13497422306680461283 c[2, 1] + 391425246893733377207 c[2, 2] + 
 
>     11351332159918267939003 c[2, 3] + 329188632637629770231087 c[2, 4] + 
 
>     9546470346491263336701523 c[2, 5] + 276847640048246636764344167 c[2, 6] + 
 
>     8028581561399152466165980843 c[2, 7] + 232828865280575421518813444447 c[2, 8] + 
 
>     6752037093136687224045589888963 c[2, 9] + 195809075700963929497322106779927 c[2, 10] + 
 
>     1832624145237590598 c[3, 0] + 53146100211890127342 c[3, 1] + 1541236906144813692918 c[3, 2] + 
 
>     44695870278199597094622 c[3, 3] + 1296180238067788315744038 c[3, 4] + 
 
>     37589226903965861156577102 c[3, 5] + 1090087580215009973540735958 c[3, 6] + 
 
>     31612539826235289232681342782 c[3, 7] + 916763654960823387747758940678 c[3, 8] + 
 
>     26586145993863878244685009279662 c[3, 9] + 770998233822052469095865269110198 c[3, 10], 
 
>    118264582638630248 c[0, 0] + 3547937479158907440 c[0, 1] + 106438124374767223200 c[0, 2] + 
 
>     3193143731243016696000 c[0, 3] + 95794311937290500880000 c[0, 4] + 
 
>     2873829358118715026400000 c[0, 5] + 86214880743561450792000000 c[0, 6] + 
 
>     2586446422306843523760000000 c[0, 7] + 77593392669205305712800000000 c[0, 8] + 
 
>     2327801780076159171384000000000 c[0, 9] + 69834053402284775141520000000000 c[0, 10] + 
 
>     465428355402774527 c[1, 0] + 13962850662083235810 c[1, 1] + 418885519862497074300 c[1, 2] + 
 
>     12566565595874912229000 c[1, 3] + 376996967876247366870000 c[1, 4] + 
 
>     11309909036287421006100000 c[1, 5] + 339297271088622630183000000 c[1, 6] + 
 
>     10178918132658678905490000000 c[1, 7] + 305367543979760367164700000000 c[1, 8] + 
 
>     9161026319392811014941000000000 c[1, 9] + 274830789581784330448230000000000 c[1, 10] + 
 
>     1832624145237590598 c[2, 0] + 54978724357127717940 c[2, 1] + 1649361730713831538200 c[2, 2] + 
 
>     49480851921414946146000 c[2, 3] + 1484425557642448384380000 c[2, 4] + 
 
>     44532766729273451531400000 c[2, 5] + 1335983001878203545942000000 c[2, 6] + 
 
>     40079490056346106378260000000 c[2, 7] + 1202384701690383191347800000000 c[2, 8] + 
 
>     36071541050711495740434000000000 c[2, 9] + 1082146231521344872213020000000000 c[2, 10] + 
 
>     7219428442606236269 c[3, 0] + 216582853278187088070 c[3, 1] + 6497485598345612642100 c[3, 2] + 
 
>     194924567950368379263000 c[3, 3] + 5847737038511051377890000 c[3, 4] + 
 
>     175432111155331541336700000 c[3, 5] + 5262963334659946240101000000 c[3, 6] + 
 
>     157888900039798387203030000000 c[3, 7] + 4736667001193951616090900000000 c[3, 8] + 
 
>     142100010035818548482727000000000 c[3, 9] + 4263000301074556454481810000000000 c[3, 10], 
 
>    465428355402774527 c[0, 0] + 14428279017486010337 c[0, 1] + 447276649542066320447 c[0, 2] + 
 
>     13865576135804055933857 c[0, 3] + 429832860209925733949567 c[0, 4] + 
 
>     13324818666507697752436577 c[0, 5] + 413069378661738630325533887 c[0, 6] + 
 
>     12805150738513897540091550497 c[0, 7] + 396959672893930823742838065407 c[0, 8] + 
 
>     12305749859711855536027980027617 c[0, 9] + 381478245651067521616867380856127 c[0, 10] + 
 
>     1832624145237590598 c[1, 0] + 56811348502365308538 c[1, 1] + 1761151803573324564678 c[1, 2] + 
 
>     54595705910773061505018 c[1, 3] + 1692466883233964906655558 c[1, 4] + 
 
>     52466473380252912106322298 c[1, 5] + 1626460674787840275295991238 c[1, 6] + 
 
>     50420280918423048534175728378 c[1, 7] + 1563028708471114504559447579718 c[1, 8] + 
 
>     48453889962604549641342874971258 c[1, 9] + 1502070588840741038881629124108998 c[1, 10] + 
 
>     7219428442606236269 c[2, 0] + 223802281720793324339 c[2, 1] + 6937870733344593054509 c[2, 2] + 
 
>     215073992733682384689779 c[2, 3] + 6667293774744153925383149 c[2, 4] + 
 
>     206686107017068771686877619 c[2, 5] + 6407269317529131922293206189 c[2, 6] + 
 
>     198625348843403089591089391859 c[2, 7] + 6157385814145495777323771147629 c[2, 8] + 
 
>     190878960238510369097036905576499 c[2, 9] + 5917247767393821442008144072871469 c[2, 10] + 
 
>     28453041492420485228 c[3, 0] + 882044286265035042068 c[3, 1] + 27343372874216086304108 c[3, 2] + 
 
>     847644559100698675427348 c[3, 3] + 26276981332121658938247788 c[3, 4] + 
 
>     814586421295771427085681428 c[3, 5] + 25252179060168914239656124268 c[3, 6] + 
 
>     782817550865236341429339852308 c[3, 7] + 24267344076822326584309535421548 c[3, 8] + 
 
>     752287666381492124113595598067988 c[3, 9] + 23320917657826255847521463540107628 c[3, 10], 
 
>    1832624145237590598 c[0, 0] + 58643972647602899136 c[0, 1] + 1876607124723292772352 c[0, 2] + 
 
>     60051427991145368715264 c[0, 3] + 1921645695716651798888448 c[0, 4] + 
 
>     61492662262932857564430336 c[0, 5] + 1967765192413851442061770752 c[0, 6] + 
 
>     62968486157243246145976664064 c[0, 7] + 2014991557031783876671253250048 c[0, 8] + 
 
>     64479729825017084053480104001536 c[0, 9] + 2063351354400546689711363328049152 c[0, 10] + 
 
>     7219428442606236269 c[1, 0] + 231021710163399560608 c[1, 1] + 7392694725228785939456 c[1, 2] + 
 
>     236566231207321150062592 c[1, 3] + 7570119398634276802002944 c[1, 4] + 
 
>     242243820756296857664094208 c[1, 5] + 7751802264201499445251014656 c[1, 6] + 
 
>     248057672454447982248032468992 c[1, 7] + 7937845518542335431937039007744 c[1, 8] + 
 
>     254011056593354733821985248247808 c[1, 9] + 8128353810987351482303527943929856 c[1, 10] + 
 
>     28453041492420485228 c[2, 0] + 910497327757455527296 c[2, 1] + 29135914488238576873472 c[2, 2] + 
 
>     932349263623634459951104 c[2, 3] + 29835176435956302718435328 c[2, 4] + 
 
>     954725645950601686989930496 c[2, 5] + 30551220670419253983677775872 c[2, 6] + 
 
>     977639061453416127477688827904 c[2, 7] + 31284449966509316079286042492928 c[2, 8] + 
 
>     1001102398928298114537153359773696 c[2, 9] + 32035276765705539665188907512758272 c[2, 10] + 
 
>     112186277851022626675 c[3, 0] + 3589960891232724053600 c[3, 1] + 
 
>     114878748519447169715200 c[3, 2] + 3676119952622309430886400 c[3, 3] + 
 
>     117635838483913901788364800 c[3, 4] + 3764346831485244857227673600 c[3, 5] + 
 
>     120459098607527835431285555200 c[3, 6] + 3854691155440890733801137766400 c[3, 7] + 
 
>     123350116974108503481636408524800 c[3, 8] + 3947203743171472111412365072793600 c[3, 9] + 
 
>     126310519781487107565195682329395200 c[3, 10], 
 
>    7219428442606236269 c[0, 0] + 238241138606005796877 c[0, 1] + 7861957573998191296941 c[0, 2] + 
 
>     259444599941940312799053 c[0, 3] + 8561671798084030322368749 c[0, 4] + 
 
>     282535169336773000638168717 c[0, 5] + 9323660588113509021059567661 c[0, 6] + 
 
>     307680799407745797694965732813 c[0, 7] + 10153466380455611323933869182829 c[0, 8] + 
 
>     335064390555035173689817683033357 c[0, 9] + 11057124888316160731763983540100781 c[0, 10] + 
 
>     28453041492420485228 c[1, 0] + 938950369249876012524 c[1, 1] + 30985362185245908413292 c[1, 2] + 
 
>     1022516952113114977638636 c[1, 3] + 33743059419732794262074988 c[1, 4] + 
 
>     1113520960851182210648474604 c[1, 5] + 36746191708089012951399661932 c[1, 6] + 
 
>     1212624326366937427396188843756 c[1, 7] + 40016602770108935104074231843948 c[1, 8] + 
 
>     1320547891413594858434449650850284 c[1, 9] + 43578080416648630328336838478059372 c[1, 10] + 
 
>     112186277851022626675 c[2, 0] + 3702147169083746680275 c[2, 1] + 
 
>     122170856579763640449075 c[2, 2] + 4031638267132200134819475 c[2, 3] + 
 
>     133044062815362604449042675 c[2, 4] + 4390454072906965946818408275 c[2, 5] + 
 
>     144884984405929876245007473075 c[2, 6] + 4781204485395685916085246611475 c[2, 7] + 
 
>     157779748018057635230813138178675 c[2, 8] + 5206731684595901962616833559896275 c[2, 9] + 
 
>     171822145591664764766355507476577075 c[2, 10] + 442512540345556302596 c[3, 0] + 
 
>     14602913831403357985668 c[3, 1] + 481896156436310813527044 c[3, 2] + 
 
>     15902573162398256846392452 c[3, 3] + 524784914359142475930950916 c[3, 4] + 
 
>     17317902173851701705721380228 c[3, 5] + 571490771737106156288805547524 c[3, 6] + 
 
>     18859195467324503157530583068292 c[3, 7] + 622353450421708604198509241253636 c[3, 8] + 
 
>     20537663863916383938550804961369988 c[3, 9] + 677742907509240669972176563725209604 c[3, 10], 
 
>    28453041492420485228 c[0, 0] + 967403410742296497752 c[0, 1] + 32891715965238080923568 c[0, 2] + 
 
>     1118318342818094751401312 c[0, 3] + 38022823655815221547644608 c[0, 4] + 
 
>     1292776004297717532619916672 c[0, 5] + 43954384146122396109077166848 c[0, 6] + 
 
>     1494449060968161467708623672832 c[0, 7] + 50811268072917489902093204876288 c[0, 8] + 
 
>     1727583114479194656671168965793792 c[0, 9] + 58737825892292618326819744836988928 c[0, 10] + 
 
>     112186277851022626675 c[1, 0] + 3814333446934769306950 c[1, 1] + 
 
>     129687337195782156436300 c[1, 2] + 4409369464656593318834200 c[1, 3] + 
 
>     149918561798324172840362800 c[1, 4] + 5097231101143021876572335200 c[1, 5] + 
 
>     173305857438862743803459396800 c[1, 6] + 5892399152921333289317619491200 c[1, 7] + 
 
>     200341571199325331836799062700800 c[1, 8] + 6811613420777061282451168131827200 c[1, 9] + 
 
>     231594856306420083603339716482124800 c[1, 10] + 442512540345556302596 c[2, 0] + 
 
>     15045426371748914288264 c[2, 1] + 511544496639463085800976 c[2, 2] + 
 
>     17392512885741744917233184 c[2, 3] + 591345438115219327185928256 c[2, 4] + 
 
>     20105744895917457124321560704 c[2, 5] + 683595326461193542226933063936 c[2, 6] + 
 
>     23242241099680580435715724173824 c[2, 7] + 790236197389139734814334621910016 c[2, 8] + 
 
>     26868030711230750983687377144940544 c[2, 9] + 913513044181845533445370822927978496 c[2, 10] + 
 
>     1746130564473065213957 c[3, 0] + 59368439192084217274538 c[3, 1] + 
 
>     2018526932530863387334292 c[3, 2] + 68629915706049355169365928 c[3, 3] + 
 
>     2333417134005678075758441552 c[3, 4] + 79336182556193054575787012768 c[3, 5] + 
 
>     2697430206910563855576758434112 c[3, 6] + 91712627034959171089609786759808 c[3, 7] + 
 
>     3118229319188611817046732749833472 c[3, 8] + 106019796852412801779588913494338048 c[3, 9] + 
 
>     3604673092982035260506023058807493632 c[3, 10], 
 
>    112186277851022626675 c[0, 0] + 3926519724785791933625 c[0, 1] + 
 
>     137428190367502717676875 c[0, 2] + 4809986662862595118690625 c[0, 3] + 
 
>     168349533200190829154171875 c[0, 4] + 5892233662006679020396015625 c[0, 5] + 
 
>     206228178170233765713860546875 c[0, 6] + 7217986235958181799985119140625 c[0, 7] + 
 
>     252629518258536362999479169921875 c[0, 8] + 8842033139048772704981770947265625 c[0, 9] + 
 
>     309471159866707044674361983154296875 c[0, 10] + 442512540345556302596 c[1, 0] + 
 
>     15487938912094470590860 c[1, 1] + 542077861923306470680100 c[1, 2] + 
 
>     18972725167315726473803500 c[1, 3] + 664045380856050426583122500 c[1, 4] + 
 
>     23241588329961764930409287500 c[1, 5] + 813455591548661772564325062500 c[1, 6] + 
 
>     28470945704203162039751377187500 c[1, 7] + 996483099647110671391298201562500 c[1, 8] + 
 
>     34876908487648873498695437054687500 c[1, 9] + 1220691797067710572454340296914062500 c[1, 10] + 
 
>     1746130564473065213957 c[2, 0] + 61114569756557282488495 c[2, 1] + 
 
>     2139009941479504887097325 c[2, 2] + 74865347951782671048406375 c[2, 3] + 
 
>     2620287178312393486694223125 c[2, 4] + 91710051240933772034297809375 c[2, 5] + 
 
>     3209851793432682021200423328125 c[2, 6] + 112344812770143870742014816484375 c[2, 7] + 
 
>     3932068446955035475970518576953125 c[2, 8] + 137622395643426241658968150193359375 c[2, 9] + 
 
>     4816783847519918458063885256767578125 c[2, 10] + 6892620648968139316416 c[3, 0] + 
 
>     241241722713884876074560 c[3, 1] + 8443460294985970662609600 c[3, 2] + 
 
>     295521110324508973191336000 c[3, 3] + 10343238861357814061696760000 c[3, 4] + 
 
>     362013360147523492159386600000 c[3, 5] + 12670467605163322225578531000000 c[3, 6] + 
 
>     443466366180716277895248585000000 c[3, 7] + 15521322816325069726333700475000000 c[3, 8] + 
 
>     543246298571377440421679516625000000 c[3, 9] + 19013620449998210414758783081875000000 c[3, 10], 
 
>    442512540345556302596 c[0, 0] + 15930451452440026893456 c[0, 1] + 
 
>     573496252287840968164416 c[0, 2] + 20645865082362274853918976 c[0, 3] + 
 
>     743251142965041894741083136 c[0, 4] + 26757041146741508210678992896 c[0, 5] + 
 
>     963253481282694295584443744256 c[0, 6] + 34677125326176994641039974793216 c[0, 7] + 
 
>     1248376511742371807077439092555776 c[0, 8] + 44941554422725385054787807332007936 c[0, 9] + 
 
>     1617895959218113861972361063952285696 c[0, 10] + 1746130564473065213957 c[1, 0] + 
 
>     62860700321030347702452 c[1, 1] + 2262985211557092517288272 c[1, 2] + 
 
>     81467467616055330622377792 c[1, 3] + 2932828834177991902405600512 c[1, 4] + 
 
>     105581838030407708486601618432 c[1, 5] + 3800946169094677505517658263552 c[1, 6] + 
 
>     136834062087408390198635697487872 c[1, 7] + 4926026235146702047150885109563392 c[1, 8] + 
 
>     177336944465281273697431863944282112 c[1, 9] + 6384130000750125853107547101994156032 c[1, 10] + 
 
>     6892620648968139316416 c[2, 0] + 248134343362853015390976 c[2, 1] + 
 
>     8932836361062708554075136 c[2, 2] + 321582108998257507946704896 c[2, 3] + 
 
>     11576955923937270286081376256 c[2, 4] + 416770413261741730298929545216 c[2, 5] + 
 
>     15003734877422702290761463627776 c[2, 6] + 540134455587217282467412690599936 c[2, 7] + 
 
>     19444840401139822168826856861597696 c[2, 8] + 700014254441033598077766847017517056 c[2, 9] + 
 
>     25200513159877209530799606492630614016 c[2, 10] + 27217014869748787888807 c[3, 0] + 
 
>     979812535310956363997052 c[3, 1] + 35273251271194429103893872 c[3, 2] + 
 
>     1269837045762999447740179392 c[3, 3] + 45714133647467980118646458112 c[3, 4] + 
 
>     1645708811308847284271272492032 c[3, 5] + 59245517207118502233765809713152 c[3, 6] + 
 
>     2132838619456266080415569149673472 c[3, 7] + 76782190300425578894960489388244992 c[3, 8] + 
 
>     2764158850815320840218577617976819712 c[3, 9] + 99509718629351550247868794247165509632 c[3, 10], 
 
>    1746130564473065213957 c[0, 0] + 64606830885503412916409 c[0, 1] + 
 
>     2390452742763626277907133 c[0, 2] + 88446751482254172282563921 c[0, 3] + 
 
>     3272529804843404374454865077 c[0, 4] + 121083602779205961854830007849 c[0, 5] + 
 
>     4480093302830620588628710290413 c[0, 6] + 165763452204732961779262280745281 c[0, 7] + 
 
>     6133247731575119585832704387575397 c[0, 8] + 226930166068279424675810062340289689 c[0, 9] + 
 
>     8396416144526338713004972306590718493 c[0, 10] + 6892620648968139316416 c[1, 0] + 
 
>     255026964011821154707392 c[1, 1] + 9435997668437382724173504 c[1, 2] + 
 
>     349131913732183160794419648 c[1, 3] + 12917880808090776949393526976 c[1, 4] + 
 
>     477961589899358747127560498112 c[1, 5] + 17684578826276273643719738430144 c[1, 6] + 
 
>     654329416572222124817630321915328 c[1, 7] + 24210188413172218618252321910867136 c[1, 8] + 
 
>     895776971287372088875335910702084032 c[1, 9] + 33143747937632767288387428695977109184 c[1, 10] + 
 
>     27217014869748787888807 c[2, 0] + 1007029550180705151885859 c[2, 1] + 
 
>     37260093356686090619776783 c[2, 2] + 1378623454197385352931740971 c[2, 3] + 
 
>     51009067805303258058474415927 c[2, 4] + 1887335508796220548163553389299 c[2, 5] + 
 
>     69831413825460160282051475404063 c[2, 6] + 2583762311542025930435904589950331 c[2, 7] + 
 
>     95599205527054959426128469828162247 c[2, 8] + 3537170604501033498766753383642003139 c[2, 9] + 
 
>     130875312366538239454369875194754116143 c[2, 10] + 107507208734435688153396 c[3, 0] + 
 
>     3977766723174120461675652 c[3, 1] + 147177368757442457081999124 c[3, 2] + 
 
>     5445562644025370912033967588 c[3, 3] + 201485817828938723745256800756 c[3, 4] + 
 
>     7454975259670732778574501627972 c[3, 5] + 275834084607817112807256560234964 c[3, 6] + 
 
>     10205861130489233173868492728693668 c[3, 7] + 377616861828101627433134230961665716 c[3, 8] + 
 
>     13971823887639760215025966545581631492 c[3, 9] + 516957483842671127955960762186520365204 c[3, 10]
 
>     , 6892620648968139316416 c[0, 0] + 261919584660789294023808 c[0, 1] + 
 
>     9952944217109993172904704 c[0, 2] + 378211880250179740570378752 c[0, 3] + 
 
>     14372051449506830141674392576 c[0, 4] + 546137955081259545383626917888 c[0, 5] + 
 
>     20753242293087862724577822879744 c[0, 6] + 788623207137338783533957269430272 c[0, 7] + 
 
>     29967681871218873774290376238350336 c[0, 8] + 1138771911106317203423034297057312768 c[0, 9] + 
 
>     43273332622040053730075303288177885184 c[0, 10] + 27217014869748787888807 c[1, 0] + 
 
>     1034246565050453939774666 c[1, 1] + 39301369471917249711437308 c[1, 2] + 
 
>     1493452039932855489034617704 c[1, 3] + 56751177517448508583315472752 c[1, 4] + 
 
>     2156544745663043326165987964576 c[1, 5] + 81948700335195646394307542653888 c[1, 6] + 
 
>     3114050612737434562983686620847744 c[1, 7] + 118333923284022513393380091592214272 c[1, 8] + 
 
>     4496689084792855508948443480504142336 c[1, 9] + 
 
>     170874185222128509340040852259157408768 c[1, 10] + 107507208734435688153396 c[2, 0] + 
 
>     4085273931908556149829048 c[2, 1] + 155240409412525133693503824 c[2, 2] + 
 
>     5899135557675955080353145312 c[2, 3] + 224167151191686293053419521856 c[2, 4] + 
 
>     8518351745284079136029941830528 c[2, 5] + 323697366320795007169137789560064 c[2, 6] + 
 
>     12300499920190210272427236003282432 c[2, 7] + 467418996967227990352234968124732416 c[2, 8] + 
 
>     17761921884754663633384928788739831808 c[2, 9] + 
 
>     674953031620677218068627293972113608704 c[2, 10] + 424784580850990744953313 c[3, 0] + 
 
>     16141814072337648308225894 c[3, 1] + 613388934748830635712583972 c[3, 2] + 
 
>     23308779520455564157078190936 c[3, 3] + 885733621777311437968971255568 c[3, 4] + 
 
>     33657877627537834642820907711584 c[3, 5] + 1278999349846437716427194493040192 c[3, 6] + 
 
>     48601975294164633224233390735527296 c[3, 7] + 1846875061178256062520868847950037248 c[3, 8] + 
 
>     70181252324773730375793016222101415424 c[3, 9] + 
 
>     2666887588341401754280134616439853786112 c[3, 10], 
 
>    27217014869748787888807 c[0, 0] + 1061463579920202727663473 c[0, 1] + 
 
>     41397079616887906378875447 c[0, 2] + 1614486105058628348776142433 c[0, 3] + 
 
>     62964958097286505602269554887 c[0, 4] + 2455633365794173718488512640593 c[0, 5] + 
 
>     95769701265972775021051992983127 c[0, 6] + 3735018349372938225821027726341953 c[0, 7] + 
 
>     145665715625544590807020081327336167 c[0, 8] + 5680962909396239041473783171766110513 c[0, 9] + 
 
>     221557553466453322617477543698878310007 c[0, 10] + 107507208734435688153396 c[1, 0] + 
 
>     4192781140642991837982444 c[1, 1] + 163518464485076681681315316 c[1, 2] + 
 
>     6377220114917990585571297324 c[1, 3] + 248711584481801632837280595636 c[1, 4] + 
 
>     9699751794790263680653943229804 c[1, 5] + 378290319996820283545503785962356 c[1, 6] + 
 
>     14753322479875991058274647652531884 c[1, 7] + 575379576715163651272711258448743476 c[1, 8] + 
 
>     22439803491891382399635739079500995564 c[1, 9] + 
 
>     875152336183763913585793824100538826996 c[1, 10] + 424784580850990744953313 c[2, 0] + 
 
>     16566598653188639053179207 c[2, 1] + 646097347474356923073989073 c[2, 2] + 
 
>     25197796551499919999885573847 c[2, 3] + 982714065508496879995537380033 c[2, 4] + 
 
>     38325848554831378319825957821287 c[2, 5] + 1494708093638423754473212355030193 c[2, 6] + 
 
>     58293615651898526424455281846177527 c[2, 7] + 2273451010424042530553755992000923553 c[2, 8] + 
 
>     88664589406537658691596483688036018567 c[2, 9] + 
 
>     3457918986854968688972262863833404724113 c[2, 10] + 1678910486216289136832512 c[3, 0] + 
 
>     65477508962435276336467968 c[3, 1] + 2553622849534975777122250752 c[3, 2] + 
 
>     99591291131864055307767779328 c[3, 3] + 3884060354142698157002943393792 c[3, 4] + 
 
>     151478353811565228123114792357888 c[3, 5] + 5907655798651043896801476901957632 c[3, 6] + 
 
>     230398576147390711975257599176347648 c[3, 7] + 8985544469748237767035046367877558272 c[3, 8] + 
 
>     350436234320181272914366808347224772608 c[3, 9] + 
 
>     13667013138487069643660305525541766131712 c[3, 10], 
 
>    107507208734435688153396 c[0, 0] + 4300288349377427526135840 c[0, 1] + 
 
>     172011533975097101045433600 c[0, 2] + 6880461359003884041817344000 c[0, 3] + 
 
>     275218454360155361672693760000 c[0, 4] + 11008738174406214466907750400000 c[0, 5] + 
 
>     440349526976248578676310016000000 c[0, 6] + 17613981079049943147052400640000000 c[0, 7] + 
 
>     704559243161997725882096025600000000 c[0, 8] + 28182369726479909035283841024000000000 c[0, 9] + 
 
>     1127294789059196361411353640960000000000 c[0, 10] + 424784580850990744953313 c[1, 0] + 
 
>     16991383234039629798132520 c[1, 1] + 679655329361585191925300800 c[1, 2] + 
 
>     27186213174463407677012032000 c[1, 3] + 1087448526978536307080481280000 c[1, 4] + 
 
>     43497941079141452283219251200000 c[1, 5] + 1739917643165658091328770048000000 c[1, 6] + 
 
>     69596705726626323653150801920000000 c[1, 7] + 2783868229065052946126032076800000000 c[1, 8] + 
 
>     111354729162602117845041283072000000000 c[1, 9] + 
 
>     4454189166504084713801651322880000000000 c[1, 10] + 1678910486216289136832512 c[2, 0] + 
 
>     67156419448651565473300480 c[2, 1] + 2686256777946062618932019200 c[2, 2] + 
 
>     107450271117842504757280768000 c[2, 3] + 4298010844713700190291230720000 c[2, 4] + 
 
>     171920433788548007611649228800000 c[2, 5] + 6876817351541920304465969152000000 c[2, 6] + 
 
>     275072694061676812178638766080000000 c[2, 7] + 11002907762467072487145550643200000000 c[2, 8] + 
 
>     440116310498682899485822025728000000000 c[2, 9] + 
 
>     17604652419947315979432881029120000000000 c[2, 10] + 6637553085032551566172515 c[3, 0] + 
 
>     265502123401302062646900600 c[3, 1] + 10620084936052082505876024000 c[3, 2] + 
 
>     424803397442083300235040960000 c[3, 3] + 16992135897683332009401638400000 c[3, 4] + 
 
>     679685435907333280376065536000000 c[3, 5] + 27187417436293331215042621440000000 c[3, 6] + 
 
>     1087496697451733248601704857600000000 c[3, 7] + 43499867898069329944068194304000000000 c[3, 8] + 
 
>     1739994715922773197762727772160000000000 c[3, 9] + 
 
>     69599788636910927910509110886400000000000 c[3, 10], 
 
>    424784580850990744953313 c[0, 0] + 17416167814890620543085833 c[0, 1] + 
 
>     714062880410515442266519153 c[0, 2] + 29276578096831133132927285273 c[0, 3] + 
 
>     1200339701970076458450018696193 c[0, 4] + 49213927780773134796450766543913 c[0, 5] + 
 
>     2017771039011698526654481428300433 c[0, 6] + 82728612599479639592833738560317753 c[0, 7] + 
 
>     3391873116578665223306183280973027873 c[0, 8] + 
 
>     139066797779725274155553514519894142793 c[0, 9] + 
 
>     5701738708968736240377694095315659854513 c[0, 10] + 1678910486216289136832512 c[1, 0] + 
 
>     68835329934867854610132992 c[1, 1] + 2822248527329582039015452672 c[1, 2] + 
 
>     115712189620512863599633559552 c[1, 3] + 4744199774441027407584975941632 c[1, 4] + 
 
>     194512190752082123710984013606912 c[1, 5] + 7974999820835367072150344557883392 c[1, 6] + 
 
>     326974992654250049958164126873219072 c[1, 7] + 13405974698824252048284729201801981952 c[1, 8] + 
 
>     549644962651794333979673897273881260032 c[1, 9] + 
 
>     22535443468723567693166629788229131661312 c[1, 10] + 6637553085032551566172515 c[2, 0] + 
 
>     272139676486334614213073115 c[2, 1] + 11157726735939719182735997715 c[2, 2] + 
 
>     457466796173528486492175906315 c[2, 3] + 18756138643114667946179212158915 c[2, 4] + 
 
>     769001684367701385793347698515515 c[2, 5] + 31529069059075756817527255639136115 c[2, 6] + 
 
>     1292691831422106029518617481204580715 c[2, 7] + 53000365088306347210263316729387809315 c[2, 8] + 
 
>     2173014968620560235620795985904900181915 c[2, 9] + 
 
>     89093613713442969660452635422100907458515 c[2, 10] + 26248505381702443375091400 c[3, 0] + 
 
>     1076188720649800178378747400 c[3, 1] + 44123737546641807313528643400 c[3, 2] + 
 
>     1809073239412314099854674379400 c[3, 3] + 74172002815904878094041649555400 c[3, 4] + 
 
>     3041052115452100001855707631771400 c[3, 5] + 124683136733536100076084012902627400 c[3, 6] + 
 
>     5112008606074980103119444529007723400 c[3, 7] + 
 
>     209592352849074184227897225689316659400 c[3, 8] + 
 
>     8593286466812041553343786253261983035400 c[3, 9] + 
 
>     352324745139293703687095236383741304451400 c[3, 10], 
 
>    1678910486216289136832512 c[0, 0] + 70514240421084143746965504 c[0, 1] + 
 
>     2961598097685534037372551168 c[0, 2] + 124387120102792429569647149056 c[0, 3] + 
 
>     5224259044317282041925180260352 c[0, 4] + 219418879861325845760857570934784 c[0, 5] + 
 
>     9215592954175685521956017979260928 c[0, 6] + 387054904075378791922152755128958976 c[0, 7] + 
 
>     16256305971165909260730415715416276992 c[0, 8] + 
 
>     682764850788968188950677460047483633664 c[0, 9] + 
 
>     28676123733136663935928453321994312613888 c[0, 10] + 6637553085032551566172515 c[1, 0] + 
 
>     278777229571367165779245630 c[1, 1] + 11708643641997420962728316460 c[1, 2] + 
 
>     491763032963891680434589291320 c[1, 3] + 20654047384483450578252750235440 c[1, 4] + 
 
>     867469990148304924286615509888480 c[1, 5] + 36433739586228806820037851415316160 c[1, 6] + 
 
>     1530217062621609886441589759443278720 c[1, 7] + 64269116630107615230546769896617706240 c[1, 8] + 
 
>     2699302898464519839682964335657943662080 c[1, 9] + 
 
>     113370721735509833266684502097633633807360 c[1, 10] + 26248505381702443375091400 c[2, 0] + 
 
>     1102437226031502621753838800 c[2, 1] + 46302363493323110113661229600 c[2, 2] + 
 
>     1944699266719570624773771643200 c[2, 3] + 81677369202221966240498409014400 c[2, 4] + 
 
>     3430449506493322582100933178604800 c[2, 5] + 144078879272719548448239193501401600 c[2, 6] + 
 
>     6051312929454221034826046127058867200 c[2, 7] + 
 
>     254155143037077283462693937336472422400 c[2, 8] + 
 
>     10674516007557245905433145368131841740800 c[2, 9] + 
 
>     448329672317404328028192105461537353113600 c[2, 10] + 103827421287588595741851077 c[3, 0] + 
 
>     4360751694078721021157745234 c[3, 1] + 183151571151306282888625299828 c[3, 2] + 
 
>     7692365988354863881322262592776 c[3, 3] + 323079371510904283015535028896592 c[3, 4] + 
 
>     13569333603457979886652471213656864 c[3, 5] + 569912011345235155239403790973588288 c[3, 6] + 
 
>     23936304476499876520054959220890708096 c[3, 7] + 
 
>     1005324788012994813842308287277409740032 c[3, 8] + 
 
>     42223641096545782181376948065651209081344 c[3, 9] + 
 
>     1773392926054922851617831818757350781416448 c[3, 10], 
 
>    6637553085032551566172515 c[0, 0] + 285414782656399717345418145 c[0, 1] + 
 
>     12272835654225187845852980235 c[0, 2] + 527731933131683077371678150105 c[0, 3] + 
 
>     22692473124662372326982160454515 c[0, 4] + 975776344360482010060232899544145 c[0, 5] + 
 
>     41958382807500726432590014680398235 c[0, 6] + 1804210460722531236601370631257124105 c[0, 7] + 
 
>     77581049811068843173858937144056336515 c[0, 8] + 
 
>     3335985141875960256475934297194422470145 c[0, 9] + 
 
>     143447361100666291028465174779360166216235 c[0, 10] + 26248505381702443375091400 c[1, 0] + 
 
>     1128685731413205065128930200 c[1, 1] + 48533486450767817800543998600 c[1, 2] + 
 
>     2086939917383016165423391939800 c[1, 3] + 89738416447469695113205853411400 c[1, 4] + 
 
>     3858751907241196889867851696690200 c[1, 5] + 165926332011371466264317622957678600 c[1, 6] + 
 
>     7134832276488973049365657787180179800 c[1, 7] + 
 
>     306797787889025841122723284848747731400 c[1, 8] + 
 
>     13192304879228111168277101248496152450200 c[1, 9] + 
 
>     567269109806808780235915353685334555358600 c[1, 10] + 103827421287588595741851077 c[2, 0] + 
 
>     4464579115366309616899596311 c[2, 1] + 191976901960751313526682641373 c[2, 2] + 
 
>     8255006784312306481647353579039 c[2, 3] + 354965291725429178710836203898677 c[2, 4] + 
 
>     15263507544193454684565956767643111 c[2, 5] + 656330824400318551436336141008653773 c[2, 6] + 
 
>     28222225449213697711762454063372112239 c[2, 7] + 
 
>     1213555694316189001605785524725000826277 c[2, 8] + 
 
>     52182894855596127069048777563175035529911 c[2, 9] + 
 
>     2243864478790633463969097435216526527786173 c[2, 10] + 410795449442129518076452040 c[3, 0] + 
 
>     17664204326011569277287437720 c[3, 1] + 759560786018497478923359821960 c[3, 2] + 
 
>     32661113798795391593704472344280 c[3, 3] + 1404427893348201838529292310804040 c[3, 4] + 
 
>     60390399413972679056759569364573720 c[3, 5] + 2596787174800825199440661482676669960 c[3, 6] + 
 
>     111661848516435483575948443755096808280 c[3, 7] + 
 
>     4801459486206725793765783081469162756040 c[3, 8] + 
 
>     206462757906889209131928672503173998509720 c[3, 9] + 
 
>     8877898589996235992672932917636481935917960 c[3, 10], 
 
>    26248505381702443375091400 c[0, 0] + 1154934236794907508504021600 c[0, 1] + 
 
>     50817106418975930374176950400 c[0, 2] + 2235952682434940936463785817600 c[0, 3] + 
 
>     98381918027137401204406575974400 c[0, 4] + 4328804393194045652993889342873600 c[0, 5] + 
 
>     190467393300538008731731131086438400 c[0, 6] + 8380565305223672384196169767803289600 c[0, 7] + 
 
>     368744873429841584904631469783344742400 c[0, 8] + 
 
>     16224774430913029735803784670467168665600 c[0, 9] + 
 
>     713890074960173308375366525500555421286400 c[0, 10] + 103827421287588595741851077 c[1, 0] + 
 
>     4568406536653898212641447388 c[1, 1] + 201009887612771521356223685072 c[1, 2] + 
 
>     8844435054961946939673842143168 c[1, 3] + 389155142418325665345649054299392 c[1, 4] + 
 
>     17122826266406329275208558389173248 c[1, 5] + 753404355721878488109176569123622912 c[1, 6] + 
 
>     33149791651762653476803769041439408128 c[1, 7] + 
 
>     1458590832677556752979365837823333957632 c[1, 8] + 
 
>     64177996637812497131092096864226694135808 c[1, 9] + 
 
>     2823831852063749873768052262025974541975552 c[1, 10] + 410795449442129518076452040 c[2, 0] + 
 
>     18074999775453698795363889760 c[2, 1] + 795299990119962746996011149440 c[2, 2] + 
 
>     34993199565278360867824490575360 c[2, 3] + 1539700780872247878184277585315840 c[2, 4] + 
 
>     67746834358378906640108213753896960 c[2, 5] + 2980860711768671892164761405171466240 c[2, 6] + 
 
>     131157871317821563255249501827544514560 c[2, 7] + 
 
>     5770946337984148783230978080411958640640 c[2, 8] + 
 
>     253921638871302546462163035538126180188160 c[2, 9] + 
 
>     11172552110337312044335173563677551928279040 c[2, 10] + 1625701140345310988037074671 c[3, 0] + 
 
>     71530850175193683473631285524 c[3, 1] + 3147357407708522072839776563056 c[3, 2] + 
 
>     138483725939174971204950168774464 c[3, 3] + 6093283941323698733017807426076416 c[3, 4] + 
 
>     268104493418242744252783526747362304 c[3, 5] + 11796597710402680747122475176883941376 c[3, 6] + 
 
>     519050299257717952873388907782893420544 c[3, 7] + 
 
>     22838213167339589926429111942447310503936 c[3, 8] + 
 
>     1004881379362941956762880925467681662173184 c[3, 9] + 
 
>     44214780691969446097566760720577993135620096 c[3, 10], 
 
>    103827421287588595741851077 c[0, 0] + 4672233957941486808383298465 c[0, 1] + 
 
>     210250528107366906377248430925 c[0, 2] + 9461273764831510786976179391625 c[0, 3] + 
 
>     425757319417417985413928072623125 c[0, 4] + 19159079373783809343626763268040625 c[0, 5] + 
 
>     862158571820271420463204347061828125 c[0, 6] + 38797135731912213920844195617782265625 c[0, 7] + 
 
>     1745871107936049626437988802800201953125 c[0, 8] + 
 
>     78564199857122233189709496126009087890625 c[0, 9] + 
 
>     3535388993570500493536927325670408955078125 c[0, 10] + 410795449442129518076452040 c[1, 0] + 
 
>     18485795224895828313440341800 c[1, 1] + 831860785120312274104815381000 c[1, 2] + 
 
>     37433735330414052334716692145000 c[1, 3] + 1684518089868632355062251146525000 c[1, 4] + 
 
>     75803314044088455977801301593625000 c[1, 5] + 3411149131983980519001058571713125000 c[1, 6] + 
 
>     153501710939279123355047635727090625000 c[1, 7] + 
 
>     6907576992267560550977143607719078125000 c[1, 8] + 
 
>     310840964652040224793971462347358515625000 c[1, 9] + 
 
>     13987843409341810115728715805631133203125000 c[1, 10] + 1625701140345310988037074671 c[2, 0] + 
 
>     73156551315538994461668360195 c[2, 1] + 3292044809199254750775076208775 c[2, 2] + 
 
>     148142016413966463784878429394875 c[2, 3] + 6666390738628490870319529322769375 c[2, 4] + 
 
>     299987583238282089164378819524621875 c[2, 5] + 13499441245722694012397046878607984375 c[2, 6] + 
 
>     607474856057521230557867109537359296875 c[2, 7] + 
 
>     27336368522588455375104019929181168359375 c[2, 8] + 
 
>     1230136583516480491879680896813152576171875 c[2, 9] + 
 
>     55356146258241622134585640356591865927734375 c[2, 10] + 6435067013866580383398424348 c[3, 0] + 
 
>     289578015623996117252929095660 c[3, 1] + 13031010703079825276381809304700 c[3, 2] + 
 
>     586395481638592137437181418711500 c[3, 3] + 26387796673736646184673163842017500 c[3, 4] + 
 
>     1187450850318149078310292372890787500 c[3, 5] + 53435288264316708523963156780085437500 c[3, 6] + 
 
>     2404587971894251883578342055103844687500 c[3, 7] + 
 
>     108206458735241334761025392479673010937500 c[3, 8] + 
 
>     4869290643085860064246142661585285492187500 c[3, 9] + 
 
>     219118078938863702891076419771337847148437500 c[3, 10], 
 
>    410795449442129518076452040 c[0, 0] + 18896590674337957831516793840 c[0, 1] + 
 
>     869243171019546060249772516640 c[0, 2] + 39985185866899118771489535765440 c[0, 3] + 
 
>     1839318549877359463488518645210240 c[0, 4] + 84608653294358535320471857679671040 c[0, 5] + 
 
>     3891998051540492624741705453264867840 c[0, 6] + 
 
>     179031910370862660738118450850183920640 c[0, 7] + 
 
>     8235467877059682393953448739108460349440 c[0, 8] + 
 
>     378831522344745390121858641998989176074240 c[0, 9] + 
 
>     17426250027858287945605497531953502099415040 c[0, 10] + 1625701140345310988037074671 c[1, 0] + 
 
>     74782252455884305449705434866 c[1, 1] + 3439983612970678050686450003836 c[1, 2] + 
 
>     158239246196651190331576700176456 c[1, 3] + 7279005325045954755252528208116976 c[1, 4] + 
 
>     334834244952113918741616297573380896 c[1, 5] + 15402375267797240262114349688375521216 c[1, 6] + 
 
>     708509262318673052057260085665273975936 c[1, 7] + 
 
>     32591426066658960394633963940602602893056 c[1, 8] + 
 
>     1499205599066312178153162341267719733080576 c[1, 9] + 
 
>     68963457557050360195045467698315107721706496 c[1, 10] + 6435067013866580383398424348 c[2, 0] + 
 
>     296013082637862697636327520008 c[2, 1] + 13616601801341684091271065920368 c[2, 2] + 
 
>     626363682861717468198469032336928 c[2, 3] + 28812729411639003537129575487498688 c[2, 4] + 
 
>     1325385552935394162707960472424939648 c[2, 5] + 60967735435028131484566181731547223808 c[2, 6] + 
 
>     2804515830011294048290044359651172295168 c[2, 7] + 
 
>     129007728180519526221342040543953925577728 c[2, 8] + 
 
>     5934355496303898206181733865021880576575488 c[2, 9] + 
 
>     272980352829979317484359757791006506522472448 c[2, 10] + 25477612258981419852683967561 c[3, 0] + 
 
>     1171970163913145313223462507806 c[3, 1] + 53910627540004684408279275359076 c[3, 2] + 
 
>     2479888866840215482780846666517496 c[3, 3] + 114074887874649912207918946659804816 c[3, 4] + 
 
>     5247444842233895961564271546351021536 c[3, 5] + 
 
>     241382462742759214231956491132146990656 c[3, 6] + 
 
>     11103593286166923854669998592078761570176 c[3, 7] + 
 
>     510765291163678497314819935235623032228096 c[3, 8] + 
 
>     23495203393529210876481717020838659482492416 c[3, 9] + 
 
>     1080779356102343700318158982958578336194651136 c[3, 10], 
 
>    1625701140345310988037074671 c[0, 0] + 76407953596229616437742509537 c[0, 1] + 
 
>     3591173819022791972573897948239 c[0, 2] + 168785169494071222710973203567233 c[0, 3] + 
 
>     7932902966221347467415740567659951 c[0, 4] + 372846439412403330968539806680017697 c[0, 5] + 
 
>     17523782652382956555521370913960831759 c[0, 6] + 
 
>     823617784661998958109504432956159092673 c[0, 7] + 
 
>     38710035879113951031146708348939477355631 c[0, 8] + 
 
>     1819371686318355698463895292400155435714657 c[0, 9] + 
 
>     85510469256962717827803078742807305478588879 c[0, 10] + 6435067013866580383398424348 c[1, 0] + 
 
>     302448149651729278019725944356 c[1, 1] + 14215063033631276066927119384732 c[1, 2] + 
 
>     668107962580669975145574611082404 c[1, 3] + 31401074241291488831842006720872988 c[1, 4] + 
 
>     1475850489340699975096574315881030436 c[1, 5] + 69364972999012898829538992846408430492 c[1, 6] + 
 
>     3260153730953606244988332663781196233124 c[1, 7] + 
 
>     153227225354819493514451635197716222956828 c[1, 8] + 
 
>     7201679591676516195179226854292662478970916 c[1, 9] + 
 
>     338478940808796261173423662151755136511633052 c[1, 10] + 25477612258981419852683967561 c[2, 0] + 
 
>     1197447776172126733076146475367 c[2, 1] + 56280045480089956454578884342249 c[2, 2] + 
 
>     2645162137564227953365207564085703 c[2, 3] + 124322620465518713808164755512028041 c[2, 4] + 
 
>     5843163161879379548983743509065317927 c[2, 5] + 
 
>     274628668608330838802235944926069942569 c[2, 6] + 
 
>     12907547424591549423705089411525287300743 c[2, 7] + 
 
>     606654728955802822914139202341688503134921 c[2, 8] + 
 
>     28512772260922732676964542510059359647341287 c[2, 9] + 
 
>     1340100296263368435817333497972789903425040489 c[2, 10] + 
 
>     100891344545565319234719464880 c[3, 0] + 4741893193641570004031814849360 c[3, 1] + 
 
>     222868980101153790189495297919920 c[3, 2] + 10474842064754228138906279002236240 c[3, 3] + 
 
>     492317577043448722528595113105103280 c[3, 4] + 23138926121042089958843970315939854160 c[3, 5] + 
 
>     1087529527688978228065666604849173145520 c[3, 6] + 
 
>     51113887801381976719086330427911137839440 c[3, 7] + 
 
>     2402352726664952905797057530111823478453680 c[3, 8] + 
 
>     112910578153252786572461703915255703487322960 c[3, 9] + 
 
>     5306797173202880968905700084017018063904179120 c[3, 10], 
 
>    6435067013866580383398424348 c[0, 0] + 308883216665595858403124368704 c[0, 1] + 
 
>     14826394399948601203349969697792 c[0, 2] + 711666931197532857760798545494016 c[0, 3] + 
 
>     34160012697481577172518330183712768 c[0, 4] + 1639680609479115704280879848818212864 c[0, 5] + 
 
>     78704669254997553805482232743274217472 c[0, 6] + 
 
>     3777824124239882582663147171677162438656 c[0, 7] + 
 
>     181335557963514363967831064240503797055488 c[0, 8] + 
 
>     8704106782248689470455891083544182258663424 c[0, 9] + 
 
>     417797125547937094581882772010120748415844352 c[0, 10] + 25477612258981419852683967561 c[1, 0] + 
 
>     1222925388431108152928830442928 c[1, 1] + 58700418644693191340583861260544 c[1, 2] + 
 
>     2817620094945273184348025340506112 c[1, 3] + 135245764557373112848705216344293376 c[1, 4] + 
 
>     6491796698753909416737850384526082048 c[1, 5] + 
 
>     311606241540187652003416818457251938304 c[1, 6] + 
 
>     14957099593929007296164007285948093038592 c[1, 7] + 
 
>     717940780508592350215872349725508465852416 c[1, 8] + 
 
>     34461157464412432810361872786824406360915968 c[1, 9] + 
 
>     1654135558291796774897369893767571505323966464 c[1, 10] + 
 
>     100891344545565319234719464880 c[2, 0] + 4842784538187135323266534314240 c[2, 1] + 
 
>     232453657832982495516793647083520 c[2, 2] + 11157775575983159784806095060008960 c[2, 3] + 
 
>     535573227647191669670692562880430080 c[2, 4] + 25707514927065200144193243018260643840 c[2, 5] + 
 
>     1233960716499129606921275664876510904320 c[2, 6] + 
 
>     59230114391958221132221231914072523407360 c[2, 7] + 
 
>     2843045490813994614346619131875481123553280 c[2, 8] + 
 
>     136466183559071741488637718330023093930557440 c[2, 9] + 
 
>     6550376810835443591454610479841108508666757120 c[2, 10] + 
 
>     399608854866746703831816258011 c[3, 0] + 19181225033603841783927180384528 c[3, 1] + 
 
>     920698801612984405628504658457344 c[3, 2] + 44193542477423251470168223605952512 c[3, 3] + 
 
>     2121290038916316070568074733085720576 c[3, 4] + 
 
>     101821921867983171387267587188114587648 c[3, 5] + 
 
>     4887452249663192226588844185029500207104 c[3, 6] + 
 
>     234597707983833226876264520881416009940992 c[3, 7] + 
 
>     11260689983223994890060697002307968477167616 c[3, 8] + 
 
>     540513119194751754722913456110782486904045568 c[3, 9] + 
 
>     25944629721348084226699845893317559371394187264 c[3, 10], 
 
>    25477612258981419852683967561 c[0, 0] + 1248403000690089572781514410489 c[0, 1] + 
 
>     61171747033814389066294206113961 c[0, 2] + 2997415604656905064248416099584089 c[0, 3] + 
 
>     146873364628188348148172388879620361 c[0, 4] + 7196794866781229059260447055101397689 c[0, 5] + 
 
>     352642948472280223903761905699968486761 c[0, 6] + 
 
>     17279504475141730971284333379298455851289 c[0, 7] + 
 
>     846695719281944817592932335585624336713161 c[0, 8] + 
 
>     41488090244815296062053684443695592498944889 c[0, 9] + 
 
>     2032916421995949507040630537741084032448299561 c[0, 10] + 
 
>     100891344545565319234719464880 c[1, 0] + 4943675882732700642501253779120 c[1, 1] + 
 
>     242240118253902331482561435176880 c[1, 2] + 11869765794441214242645510323667120 c[1, 3] + 
 
>     581618523927619497889630005859688880 c[1, 4] + 28499307672453355396591870287124755120 c[1, 5] + 
 
>     1396466075950214414433001644069113000880 c[1, 6] + 
 
>     68426837721560506307217080559386537043120 c[1, 7] + 
 
>     3352915048356464809053636947409940315112880 c[1, 8] + 
 
>     164292837369466775643628210423087075440531120 c[1, 9] + 
 
>     8050349031103872006537782310731266696586024880 c[1, 10] + 
 
>     399608854866746703831816258011 c[2, 0] + 19580833888470588487758996642539 c[2, 1] + 
 
>     959460860535058835900190835484411 c[2, 2] + 47013582166217882959109350938736139 c[2, 3] + 
 
>     2303665526144676264996358195998070811 c[2, 4] + 
 
>     112879610781089136984821551603905469739 c[2, 5] + 
 
>     5531100928273367712256256028591368017211 c[2, 6] + 
 
>     271023945485395017900556545400977032843339 c[2, 7] + 
 
>     13280173328784355877127270724647874609323611 c[2, 8] + 
 
>     650728493110433437979236265507745855856856939 c[2, 9] + 
 
>     31885696162411238460982577009879546936985990011 c[2, 10] + 
 
>     1583065848125953678957175639240 c[3, 0] + 77570226558171730268901606322760 c[3, 1] + 
 
>     3800941101350414783176178709815240 c[3, 2] + 186246113966170324375632756780946760 c[3, 3] + 
 
>     9126059584342345894406005082266391240 c[3, 4] + 
 
>     447176919632774948825894249031053170760 c[3, 5] + 
 
>     21911669062005972492468818202521605367240 c[3, 6] + 
 
>     1073671784038292652130972091923558662994760 c[3, 7] + 
 
>     52609917417876339954417632504254374486743240 c[3, 8] + 
 
>     2577885953475940657766463992708464349850418760 c[3, 9] + 
 
>     126316411720321092230556735642714753142670519240 c[3, 10], 
 
>    100891344545565319234719464880 c[0, 0] + 5044567227278265961735973244000 c[0, 1] + 
 
>     252228361363913298086798662200000 c[0, 2] + 12611418068195664904339933110000000 c[0, 3] + 
 
>     630570903409783245216996655500000000 c[0, 4] + 31528545170489162260849832775000000000 c[0, 5] + 
 
>     1576427258524458113042491638750000000000 c[0, 6] + 
 
>     78821362926222905652124581937500000000000 c[0, 7] + 
 
>     3941068146311145282606229096875000000000000 c[0, 8] + 
 
>     197053407315557264130311454843750000000000000 c[0, 9] + 
 
>     9852670365777863206515572742187500000000000000 c[0, 10] + 
 
>     399608854866746703831816258011 c[1, 0] + 19980442743337335191590812900550 c[1, 1] + 
 
>     999022137166866759579540645027500 c[1, 2] + 49951106858343337978977032251375000 c[1, 3] + 
 
>     2497555342917166898948851612568750000 c[1, 4] + 
 
>     124877767145858344947442580628437500000 c[1, 5] + 
 
>     6243888357292917247372129031421875000000 c[1, 6] + 
 
>     312194417864645862368606451571093750000000 c[1, 7] + 
 
>     15609720893232293118430322578554687500000000 c[1, 8] + 
 
>     780486044661614655921516128927734375000000000 c[1, 9] + 
 
>     39024302233080732796075806446386718750000000000 c[1, 10] + 
 
>     1583065848125953678957175639240 c[2, 0] + 79153292406297683947858781962000 c[2, 1] + 
 
>     3957664620314884197392939098100000 c[2, 2] + 197883231015744209869646954905000000 c[2, 3] + 
 
>     9894161550787210493482347745250000000 c[2, 4] + 
 
>     494708077539360524674117387262500000000 c[2, 5] + 
 
>     24735403876968026233705869363125000000000 c[2, 6] + 
 
>     1236770193848401311685293468156250000000000 c[2, 7] + 
 
>     61838509692420065584264673407812500000000000 c[2, 8] + 
 
>     3091925484621003279213233670390625000000000000 c[2, 9] + 
 
>     154596274231050163960661683519531250000000000000 c[2, 10] + 
 
>     6272525058612260456729162567389 c[3, 0] + 313626252930613022836458128369450 c[3, 1] + 
 
>     15681312646530651141822906418472500 c[3, 2] + 784065632326532557091145320923625000 c[3, 3] + 
 
>     39203281616326627854557266046181250000 c[3, 4] + 
 
>     1960164080816331392727863302309062500000 c[3, 5] + 
 
>     98008204040816569636393165115453125000000 c[3, 6] + 
 
>     4900410202040828481819658255772656250000000 c[3, 7] + 
 
>     245020510102041424090982912788632812500000000 c[3, 8] + 
 
>     12251025505102071204549145639431640625000000000 c[3, 9] + 
 
>     612551275255103560227457281971582031250000000000 c[3, 10]}

In[168]:= Solve[Thread[%==0]]

Out[168]= {{c[0, 5] -> 
 
       14310068122 c[0, 0]   174557566 c[0, 1]   2016793 c[0, 2]   21061 c[0, 3]   181 c[0, 4]
>      ------------------- - ----------------- + --------------- - ------------- + -----------, 
            90224199              2313441             59319            1521            39
 
                 12391484738 c[0, 0]   147733772 c[0, 1]   1628051 c[0, 2]   15344 c[0, 3]
>     c[0, 6] -> ------------------- - ----------------- + --------------- - ------------- + 
                      30074733              771147              19773             507
 
        100 c[0, 4]
>       -----------, c[0, 7] -> 
            13
 
       33301198142 c[0, 0]   389559374 c[0, 1]   4149656 c[0, 2]   36773 c[0, 3]   212 c[0, 4]
>      ------------------- - ----------------- + --------------- - ------------- + -----------, 
            90224199              2313441             59319            1521            39
 
                 8301791279 c[0, 0]   95000150 c[0, 1]   974378 c[0, 2]   8084 c[0, 3]   41 c[0, 4]
>     c[0, 8] -> ------------------ - ---------------- + -------------- - ------------ + ----------, 
                      90224199            2313441            59319            1521           39
 
                 -2495549060 c[0, 0]   29357207 c[0, 1]   315683 c[0, 2]   2843 c[0, 3]   17 c[0, 4]
>     c[0, 9] -> ------------------- + ---------------- - -------------- + ------------ - ----------, 
                      90224199             2313441            59319            1521           39
 
                  -349115132 c[0, 0]   4033586 c[0, 1]   42122 c[0, 2]   362 c[0, 3]   2 c[0, 4]
>     c[0, 10] -> ------------------ + --------------- - ------------- + ----------- - ---------, 
                       30074733            771147            19773           507          13
 
                 -21 c[0, 0]             191 c[0, 0]   21 c[0, 1]
>     c[1, 0] -> -----------, c[1, 1] -> ----------- - ----------, 
                     26                      507           26
 
                 -123079 c[0, 0]   191 c[0, 1]   21 c[0, 2]
>     c[1, 2] -> --------------- + ----------- - ----------, 
                      39546            507           26
 
                 12399223 c[0, 0]   123079 c[0, 1]   191 c[0, 2]   21 c[0, 3]
>     c[1, 3] -> ---------------- - -------------- + ----------- - ----------, 
                     1542294            39546            507           26
 
                 -1048303849 c[0, 0]   12399223 c[0, 1]   123079 c[0, 2]   191 c[0, 3]   21 c[0, 4]
>     c[1, 4] -> ------------------- + ---------------- - -------------- + ----------- - ----------, 
                      60149466             1542294            39546            507           26
 
                 -16704258275 c[0, 0]   201338849 c[0, 1]   1152055 c[0, 2]   12277 c[0, 3]
>     c[1, 5] -> -------------------- + ----------------- - --------------- + ------------- - 
                      180448398              4626882             59319            1521
 
        263 c[0, 4]
>       -----------, c[1, 6] -> 
            78
 
       -20691818617 c[0, 0]   249610099 c[0, 1]   1406315 c[0, 2]   27649 c[0, 3]   197 c[0, 4]
>      -------------------- + ----------------- - --------------- + ------------- - -----------, 
             60149466              1542294             19773            1014            26
 
                 -89412038929 c[0, 0]   1053540145 c[0, 1]   5680697 c[0, 2]   102799 c[0, 3]
>     c[1, 7] -> -------------------- + ------------------ - --------------- + -------------- - 
                      180448398              4626882              59319             3042
 
        308 c[0, 4]
>       -----------, c[1, 8] -> 
            39
 
       -20110000496 c[0, 0]   464497993 c[0, 1]   4841137 c[0, 2]   20669 c[0, 3]   223 c[0, 4]
>      -------------------- + ----------------- - --------------- + ------------- - -----------, 
             90224199              4626882            118638            1521            78
 
                 2712185683 c[0, 0]   16585433 c[0, 1]   189143 c[0, 2]   1853 c[0, 3]   25 c[0, 4]
>     c[1, 9] -> ------------------ - ---------------- + -------------- - ------------ + ----------, 
                     180448398            2313441            59319            1521           78
 
                  610951481 c[0, 0]   14117551 c[0, 1]   147427 c[0, 2]   1267 c[0, 3]   7 c[0, 4]
>     c[1, 10] -> ----------------- - ---------------- + -------------- - ------------ + ---------, 
                      30074733            1542294            39546            1014          26
 
                 3 c[0, 0]             -527 c[0, 0]   3 c[0, 1]
>     c[2, 0] -> ---------, c[2, 1] -> ------------ + ---------, 
                    13                     1014          13
 
                 228331 c[0, 0]   527 c[0, 1]   3 c[0, 2]
>     c[2, 2] -> -------------- - ----------- + ---------, 
                     79092           1014          13
 
                 -21871747 c[0, 0]   228331 c[0, 1]   527 c[0, 2]   3 c[0, 3]
>     c[2, 3] -> ----------------- + -------------- - ----------- + ---------, 
                      3084588            79092           1014          13
 
                 1809287245 c[0, 0]   21871747 c[0, 1]   228331 c[0, 2]   527 c[0, 3]   3 c[0, 4]
>     c[2, 4] -> ------------------ - ---------------- + -------------- - ----------- + ---------, 
                     120298932            3084588            79092           1014          13
 
                 2249852639 c[0, 0]   21954119 c[0, 1]   179213 c[0, 2]   1877 c[0, 3]   43 c[0, 4]
>     c[2, 5] -> ------------------ - ---------------- + -------------- - ------------ + ----------, 
                     360896796            9253764            237276           6084           78
 
                 676246343 c[0, 0]   8390597 c[0, 1]   99599 c[0, 2]   1073 c[0, 3]   9 c[0, 4]
>     c[2, 6] -> ----------------- - --------------- + ------------- - ------------ + ---------, 
                      9253764            237276            6084            156            4
 
                 75407690407 c[0, 0]   894771751 c[0, 1]   2440135 c[0, 2]   44975 c[0, 3]
>     c[2, 7] -> ------------------- - ----------------- + --------------- - ------------- + 
                      360896796             9253764             59319            3042
 
        139 c[0, 4]
>       -----------, c[2, 8] -> 
            39
 
       47942612953 c[0, 0]   555033751 c[0, 1]   5809363 c[0, 2]   12496 c[0, 3]   137 c[0, 4]
>      ------------------- - ----------------- + --------------- - ------------- + -----------, 
            360896796             9253764            237276            1521            78
 
                 -93822193 c[0, 0]   2918971 c[0, 1]   62371 c[0, 2]   991 c[0, 3]   5 c[0, 4]
>     c[2, 9] -> ----------------- + --------------- - ------------- + ----------- - ---------, 
                     360896796           9253764          237276          6084          78
 
                  -610951481 c[0, 0]   14117551 c[0, 1]   147427 c[0, 2]   1267 c[0, 3]   7 c[0, 4]
>     c[2, 10] -> ------------------ + ---------------- - -------------- + ------------ - ---------, 
                       60149466            3084588            79092            2028          52
 
                 -c[0, 0]             28 c[0, 0]   c[0, 1]
>     c[3, 0] -> --------, c[3, 1] -> ---------- - -------, 
                    26                   169         26
 
                 -8771 c[0, 0]   28 c[0, 1]   c[0, 2]
>     c[3, 2] -> ------------- + ---------- - -------, 
                     13182          169         26
 
                 789377 c[0, 0]   8771 c[0, 1]   28 c[0, 2]   c[0, 3]
>     c[3, 3] -> -------------- - ------------ + ---------- - -------, 
                     514098          13182          169         26
 
                 -63415283 c[0, 0]   789377 c[0, 1]   8771 c[0, 2]   28 c[0, 3]   c[0, 4]
>     c[3, 4] -> ----------------- + -------------- - ------------ + ---------- - -------, 
                     20049822            514098          13182          169         26
 
                 72168757 c[0, 0]   1206791 c[0, 1]   13513 c[0, 2]   202 c[0, 3]   c[0, 4]
>     c[3, 5] -> ---------------- - --------------- + ------------- - ----------- - -------, 
                    360896796           4626882           59319          1521         78
 
                 -40905715 c[0, 0]   1200239 c[0, 1]   4592 c[0, 2]   137 c[0, 3]   5 c[0, 4]
>     c[3, 6] -> ----------------- + --------------- - ------------ + ----------- - ---------, 
                     20049822            1028196           6591           338          26
 
                 -4824212405 c[0, 0]   57697745 c[0, 1]   1274401 c[0, 2]   5981 c[0, 3]   19 c[0, 4]
>     c[3, 7] -> ------------------- + ---------------- - --------------- + ------------ - ----------, 
                      180448398            4626882            237276            3042           39
 
                 -2003050405 c[0, 0]   46383977 c[0, 1]   485651 c[0, 2]   8365 c[0, 3]   23 c[0, 4]
>     c[3, 8] -> ------------------- + ---------------- - -------------- + ------------ - ----------, 
                      90224199             4626882            118638           6084           78
 
                 -61407215 c[0, 0]   111836 c[0, 1]   29 c[0, 2]   16 c[0, 3]   c[0, 4]
>     c[3, 9] -> ----------------- + -------------- - ---------- - ---------- + -------, 
                     360896796          2313441         59319         1521        156
 
                  87278783 c[0, 0]   2016793 c[0, 1]   21061 c[0, 2]   181 c[0, 3]   c[0, 4]
>     c[3, 10] -> ---------------- - --------------- + ------------- - ----------- + -------}}
                      60149466           3084588           79092          2028         52

In[169]:= ansatz /. First[%] /. c[u_,v_] -> u+v

                                                                 5                  6
                      2           3           4        70402099 n  f[n]   67032866 n  f[n]
Out[169]= n f[n] + 2 n  f[n] + 3 n  f[n] + 4 n  f[n] - ---------------- - ---------------- - 
                                                           2313441             771147
 
                7                  8                  9                 10
     183378893 n  f[n]   46157642 n  f[n]   13672850 n  f[n]   1925324 n   f[n]   21 n f[1 + n]
>    ----------------- - ---------------- + ---------------- + ---------------- - ------------- - 
          2313441            2313441            2313441             771147             26
 
          2                   3                    4                      5
     628 n  f[1 + n]   94553 n  f[1 + n]   440669 n  f[1 + n]   71254583 n  f[1 + n]
>    --------------- - ----------------- - ------------------ + -------------------- + 
           507               19773              1542294               4626882
 
               6                       7                       8                     9
     54821987 n  f[1 + n]   245130617 n  f[1 + n]   222602053 n  f[1 + n]   7321568 n  f[1 + n]
>    -------------------- + --------------------- + --------------------- - ------------------- - 
            771147                 2313441                 4626882                2313441
 
              10                               2                    3                     4
     3369317 n   f[1 + n]   3 n f[2 + n]   59 n  f[2 + n]   200875 n  f[2 + n]   6024019 n  f[2 + n]
>    -------------------- + ------------ - -------------- + ------------------ - ------------------- + 
            771147               13             1014              79092                3084588
 
             5                     6                       7                       8
     966370 n  f[2 + n]   1691245 n  f[2 + n]   411966025 n  f[2 + n]   264966805 n  f[2 + n]
>    ------------------ - ------------------- - --------------------- - --------------------- + 
          2313441               118638                 9253764                 9253764
 
             9                     10                             2                  3
     101603 n  f[2 + n]   3369317 n   f[2 + n]   n f[3 + n]   15 n  f[3 + n]   2962 n  f[3 + n]
>    ------------------ + -------------------- - ---------- + -------------- - ---------------- + 
          4626882               1542294              26            169               6591
 
             4                     5                    6                      7
     281675 n  f[3 + n]   1179491 n  f[3 + n]   226877 n  f[3 + n]   26270921 n  f[3 + n]
>    ------------------ - ------------------- + ------------------ + -------------------- + 
           514098               4626882              1028196               4626882
 
               8                   9                    10
     44261197 n  f[3 + n]   95885 n  f[3 + n]   481331 n   f[3 + n]
>    -------------------- + ----------------- - -------------------
           9253764               2313441              1542294

In[170]:= Collect[Numerator[Together[%]], f[_], Factor]

                                                     2           3
Out[170]= 4 n (1 + 2 n) (-59319 + 156663 n - 448734 n  + 962662 n ) 
 
                         2       3      4      5
>     (-39 - 103 n - 94 n  - 24 n  + 7 n  + 3 n ) f[n] - 
 
                                      2           3
>    2 n (-59319 + 156663 n - 448734 n  + 962662 n ) 
 
                          2        3        4       5       6
>     (-63 - 263 n - 591 n  - 616 n  - 223 n  + 25 n  + 21 n ) f[1 + n] + 
 
                                    2           3
>    n (-59319 + 156663 n - 448734 n  + 962662 n ) 
 
                         2        3        4       5       6
>     (-36 - 86 n - 351 n  - 556 n  - 274 n  + 10 n  + 21 n ) f[2 + n] - 
 
                                            2           3            2       3      4      5
>    n (3 + n) (-59319 + 156663 n - 448734 n  + 962662 n ) (-2 - 10 n  - 22 n  - 8 n  + 3 n ) f[3 + n]

In[171]:= ansatz = Sum[c[i,j]n^j f[n+i], {i, 0, 3}, {j, 0, 6}];

In[172]:= Collect[Numerator[Together[ansatz /. Solve[Thread[(Table[ansatz, {n, 0, 50}] /. f -> a)==0]] /. c[u_,v_] -> u+v]], f[_], Factor]

Out[172]= {0}

In[173]:= ansatz = Sum[c[i,j]n^j f[n+i], {i, 0, 3}, {j, 0, 7}];

In[174]:= Collect[Numerator[Together[ansatz /. Solve[Thread[(Table[ansatz, {n, 0, 50}] /. f -> a)==0]] /. c[u_,v_] -> u+v]], f[_], Factor]

                                             2       3      4      5
Out[174]= {-4 n (1 + 2 n) (-39 - 103 n - 94 n  - 24 n  + 7 n  + 3 n ) f[n] + 
 
                              2        3        4       5       6
>     2 n (-63 - 263 n - 591 n  - 616 n  - 223 n  + 25 n  + 21 n ) f[1 + n] - 
 
                           2        3        4       5       6
>     n (-36 - 86 n - 351 n  - 556 n  - 274 n  + 10 n  + 21 n ) f[2 + n] + 
 
                          2       3      4      5
>     n (3 + n) (-2 - 10 n  - 22 n  - 8 n  + 3 n ) f[3 + n]}

In[175]:= ansatz = Sum[c[i,j]n^j f[n+i], {i, 0, 3}, {j, 0, 6}];

In[176]:= Collect[Numerator[Together[ansatz /. Solve[Thread[(Table[ansatz, {n, 0, 50}] /. f -> a)==0]] /. c[u_,v_] -> u+v]], f[_], Factor]

Out[176]= {0}

In[177]:= Collect[Numerator[Together[ansatz /. Solve[Thread[(Table[ansatz, {n, 0, 50}] /. f -> a)==0]] /. c[u_,v_] -> u-v]], f[_], Factor]

Out[177]= {0}

In[178]:= Collect[Numerator[Together[ansatz /. Solve[Thread[(Table[ansatz, {n, 0, 50}] /. f -> a)==0]] /. c[u_,v_] -> 1]], f[_], Factor]

                                           2       3      4      5
Out[178]= {-4 (1 + 2 n) (-39 - 103 n - 94 n  - 24 n  + 7 n  + 3 n ) f[n] + 
 
                            2        3        4       5       6
>     2 (-63 - 263 n - 591 n  - 616 n  - 223 n  + 25 n  + 21 n ) f[1 + n] + 
 
                        2        3        4       5       6
>     (36 + 86 n + 351 n  + 556 n  + 274 n  - 10 n  - 21 n ) f[2 + n] + 
 
                        2       3      4      5
>     (3 + n) (-2 - 10 n  - 22 n  - 8 n  + 3 n ) f[3 + n]}

In[179]:= Collect[Numerator[Together[ansatz /. Solve[Thread[(Table[ansatz, {n, 0, 50}] /. f -> a)==0]] /. c[u_,v_] -> 1]], f[_], Factor]

                                           2       3      4      5
Out[179]= {-4 (1 + 2 n) (-39 - 103 n - 94 n  - 24 n  + 7 n  + 3 n ) f[n] + 
 
                            2        3        4       5       6
>     2 (-63 - 263 n - 591 n  - 616 n  - 223 n  + 25 n  + 21 n ) f[1 + n] + 
 
                        2        3        4       5       6
>     (36 + 86 n + 351 n  + 556 n  + 274 n  - 10 n  - 21 n ) f[2 + n] + 
 
                        2       3      4      5
>     (3 + n) (-2 - 10 n  - 22 n  - 8 n  + 3 n ) f[3 + n]}

In[180]:= ansatz = Sum[c[i,j]x^j D[f[x], {x, i}], {i, 0, 3}, {j, 0, 6}];

In[181]:= ansatz

                                           2                 3                 4
Out[181]= c[0, 0] f[x] + x c[0, 1] f[x] + x  c[0, 2] f[x] + x  c[0, 3] f[x] + x  c[0, 4] f[x] + 
 
      5                 6                                                   2
>    x  c[0, 5] f[x] + x  c[0, 6] f[x] + c[1, 0] f'[x] + x c[1, 1] f'[x] + x  c[1, 2] f'[x] + 
 
      3                  4                  5                  6
>    x  c[1, 3] f'[x] + x  c[1, 4] f'[x] + x  c[1, 5] f'[x] + x  c[1, 6] f'[x] + c[2, 0] f''[x] + 
 
                         2                   3                   4
>    x c[2, 1] f''[x] + x  c[2, 2] f''[x] + x  c[2, 3] f''[x] + x  c[2, 4] f''[x] + 
 
      5                   6                           (3)                 (3)
>    x  c[2, 5] f''[x] + x  c[2, 6] f''[x] + c[3, 0] f   [x] + x c[3, 1] f   [x] + 
 
      2          (3)       3          (3)       4          (3)       5          (3)
>    x  c[3, 2] f   [x] + x  c[3, 3] f   [x] + x  c[3, 4] f   [x] + x  c[3, 5] f   [x] + 
 
      6          (3)
>    x  c[3, 6] f   [x]

In[182]:= Collect[%, _[x]]

                                  2            3            4            5            6
Out[182]= (c[0, 0] + x c[0, 1] + x  c[0, 2] + x  c[0, 3] + x  c[0, 4] + x  c[0, 5] + x  c[0, 6]) 
 
                                     2            3            4            5            6
>     f[x] + (c[1, 0] + x c[1, 1] + x  c[1, 2] + x  c[1, 3] + x  c[1, 4] + x  c[1, 5] + x  c[1, 6]) 
 
                                      2            3            4            5            6
>     f'[x] + (c[2, 0] + x c[2, 1] + x  c[2, 2] + x  c[2, 3] + x  c[2, 4] + x  c[2, 5] + x  c[2, 6]) 
 
                                       2            3            4            5            6
>     f''[x] + (c[3, 0] + x c[3, 1] + x  c[3, 2] + x  c[3, 3] + x  c[3, 4] + x  c[3, 5] + x  c[3, 6]) 
 
       (3)
>     f   [x]

In[183]:= ansatz /. f -> (Function[x, Sum[a[n]x^n, {n, 0, 50}]+O[x]^51])
*** output flushed ***
In[184]:= Solve[Thread[CoefficientList[%, x]==0]]

                       557841739377 c[0, 0]   193807898783 c[0, 1]
Out[184]= {{c[0, 2] -> -------------------- + --------------------, 
                            1567640141             1567640141
 
                 724899967040 c[0, 0]   757710894992 c[0, 1]
>     c[0, 3] -> -------------------- + --------------------, 
                      1567640141             4702920423
 
                 5358174703 c[0, 0]   5591889349 c[0, 1]
>     c[0, 4] -> ------------------ + ------------------, c[0, 5] -> 0, c[0, 6] -> 0, 
                     1567640141           4702920423
 
                 41018548161 c[0, 0]   16162682455 c[0, 1]
>     c[1, 0] -> ------------------- + -------------------, 
                     6270560564            6270560564
 
                 256283563634 c[0, 0]   174054757403 c[0, 1]
>     c[1, 1] -> -------------------- + --------------------, 
                      1567640141             3135280282
 
                 -5862732424893 c[0, 0]   2031102185941 c[0, 1]
>     c[1, 2] -> ---------------------- - ---------------------, 
                       6270560564              6270560564
 
                 519347300023 c[0, 0]   540392719379 c[0, 1]
>     c[1, 3] -> -------------------- + --------------------, 
                      1567640141             4702920423
 
                 7482141377399 c[0, 0]   7827985850053 c[0, 1]
>     c[1, 4] -> --------------------- + ---------------------, 
                      3135280282              9405840846
 
                 26790873515 c[0, 0]   27959446745 c[0, 1]
>     c[1, 5] -> ------------------- + -------------------, c[1, 6] -> 0, 
                     1567640141            4702920423
 
                 16808129669 c[0, 0]   16162682455 c[0, 1]
>     c[2, 0] -> ------------------- + -------------------, 
                     12541121128           37623363384
 
                 -405060977861 c[0, 0]   419261573009 c[0, 1]
>     c[2, 1] -> --------------------- - --------------------, 
                      6270560564             18811681692
 
                 4412750203757 c[0, 0]   1532681933405 c[0, 1]
>     c[2, 2] -> --------------------- + ---------------------, 
                      12541121128             12541121128
 
                 -2174385804767 c[0, 0]   2266044113359 c[0, 1]
>     c[2, 3] -> ---------------------- - ---------------------, 
                       6270560564              18811681692
 
                 -6339815827515 c[0, 0]   6642555524041 c[0, 1]
>     c[2, 4] -> ---------------------- - ---------------------, 
                       6270560564              18811681692
 
                 2242185848850 c[0, 0]   782681570852 c[0, 1]
>     c[2, 5] -> --------------------- + --------------------, 
                      1567640141              1567640141
 
                 10716349406 c[0, 0]   11183778698 c[0, 1]
>     c[2, 6] -> ------------------- + -------------------, 
                     1567640141            4702920423
 
                 -9457821745 c[0, 0]   3232536491 c[0, 1]
>     c[3, 0] -> ------------------- - ------------------, 
                     37623363384          37623363384
 
                 197270091733 c[0, 0]   68123054039 c[0, 1]
>     c[3, 1] -> -------------------- + -------------------, 
                     37623363384            37623363384
 
                 -540919066081 c[0, 0]   15603147639 c[0, 1]
>     c[3, 2] -> --------------------- - -------------------, 
                      18811681692            1567640141
 
                 425232678363 c[0, 0]   438093359791 c[0, 1]
>     c[3, 3] -> -------------------- + --------------------, 
                     12541121128            37623363384
 
                 4514626117427 c[0, 0]   1585298377819 c[0, 1]
>     c[3, 4] -> --------------------- + ---------------------, 
                      37623363384             37623363384
 
                 -5882861344631 c[0, 0]   685596729705 c[0, 1]
>     c[3, 5] -> ---------------------- - --------------------, 
                      18811681692              6270560564
 
                 37333828255 c[0, 0]   13042248187 c[0, 1]
>     c[3, 6] -> ------------------- + -------------------}}
                      204474801             204474801

In[185]:= Collect[Numerator[Together[ansatz /. % /. c[0,0] -> 1 /. c[__] -> 0]], _[x], Factor]

                                          2                 3               4
Out[185]= {24 (1567640141 + 557841739377 x  + 724899967040 x  + 5358174703 x ) f[x] + 
 
                                                        2                  3                   4
>     6 (41018548161 + 1025134254536 x - 5862732424893 x  + 2077389200092 x  + 14964282754798 x  + 
 
                       5                                                           2
>        107163494060 x ) f'[x] + 3 (16808129669 - 810121955722 x + 4412750203757 x  - 
 
                        3                   4                   5                6
>        4348771609534 x  - 12679631655030 x  + 17937486790800 x  + 85730795248 x ) f''[x] + 
 
                                                                                 2                 3
>     (-1 + x) (-1 + 2 x) (-1 + 4 x) (9457821745 - 131065339518 x + 31971251106 x  + 858678049865 x ) 
 
        (3)
>      f   [x]}

In[186]:= rec1 = Sum[RandomInteger[10]n^j f[n+i], {i,0,2},{j,0,1}]

Out[186]= 4 f[n] + 10 f[1 + n] + 3 n f[1 + n] + 8 f[2 + n] + 9 n f[2 + n]

In[187]:= rec2 = Sum[RandomInteger[10]n^j f[n+i], {i,0,2},{j,0,1}]

Out[187]= 2 f[n] + 2 n f[n] + 3 f[1 + n] + 5 n f[1 + n] + 2 f[2 + n] + 3 n f[2 + n]

In[188]:= Collect[rec1, f[_], Factor]

Out[188]= 4 f[n] + (10 + 3 n) f[1 + n] + (8 + 9 n) f[2 + n]

In[189]:= Collect[rec2, f[_], Factor]

Out[189]= 2 (1 + n) f[n] + (3 + 5 n) f[1 + n] + (2 + 3 n) f[2 + n]

In[190]:= rec2 = rec2 /. f -> g;

In[191]:= ansatz = Sum[p[i]h[n+i], {i, 0, 4}]

Out[191]= h[n] p[0] + h[1 + n] p[1] + h[2 + n] p[2] + h[3 + n] p[3] + h[4 + n] p[4]

In[192]:= % /. h[n+i_.] -> f[n+i]+g[n+i]

Out[192]= (f[n] + g[n]) p[0] + (f[1 + n] + g[1 + n]) p[1] + (f[2 + n] + g[2 + n]) p[2] + 
 
>    (f[3 + n] + g[3 + n]) p[3] + (f[4 + n] + g[4 + n]) p[4]

In[193]:= rec1

Out[193]= 4 f[n] + 10 f[1 + n] + 3 n f[1 + n] + 8 f[2 + n] + 9 n f[2 + n]

In[194]:= Solve[rec1 == 0, f[n+2]]

                        -4 f[n] - 10 f[1 + n] - 3 n f[1 + n]
Out[194]= {{f[2 + n] -> ------------------------------------}}
                                      8 + 9 n

In[195]:= % /. n -> n - 2

                    -4 f[-2 + n] - 10 f[-1 + n] - 3 (-2 + n) f[-1 + n]
Out[195]= {{f[n] -> --------------------------------------------------}}
                                      8 + 9 (-2 + n)

In[196]:= % /. n -> n + i

                        -4 f[-2 + i + n] - 10 f[-1 + i + n] - 3 (-2 + i + n) f[-1 + i + n]
Out[196]= {{f[i + n] -> ------------------------------------------------------------------}}
                                                8 + 9 (-2 + i + n)

In[197]:= %192

Out[197]= (f[n] + g[n]) p[0] + (f[1 + n] + g[1 + n]) p[1] + (f[2 + n] + g[2 + n]) p[2] + 
 
>    (f[3 + n] + g[3 + n]) p[3] + (f[4 + n] + g[4 + n]) p[4]

In[198]:= %192 /. First[%196 /. i -> 4] /. First[%196 /. i -> 3] /. First[%196 /. i -> 2]

Out[198]= (f[n] + g[n]) p[0] + (f[1 + n] + g[1 + n]) p[1] + 
 
      -4 f[n] - 10 f[1 + n] - 3 n f[1 + n]
>    (------------------------------------ + g[2 + n]) p[2] + 
                    8 + 9 n
 
                     10 (-4 f[n] - 10 f[1 + n] - 3 n f[1 + n])
>    ((-4 f[1 + n] - ----------------------------------------- - 
                                      8 + 9 n
 
           3 (1 + n) (-4 f[n] - 10 f[1 + n] - 3 n f[1 + n])
>          ------------------------------------------------) / (8 + 9 (1 + n)) + g[3 + n]) p[3] + 
                               8 + 9 n
 
       -4 (-4 f[n] - 10 f[1 + n] - 3 n f[1 + n])
>    ((----------------------------------------- - 
                        8 + 9 n
 
                              10 (-4 f[n] - 10 f[1 + n] - 3 n f[1 + n])
>          (10 (-4 f[1 + n] - ----------------------------------------- - 
                                               8 + 9 n
 
                3 (1 + n) (-4 f[n] - 10 f[1 + n] - 3 n f[1 + n])
>               ------------------------------------------------)) / (8 + 9 (1 + n)) - 
                                    8 + 9 n
 
                                     10 (-4 f[n] - 10 f[1 + n] - 3 n f[1 + n])
>          (3 (2 + n) (-4 f[1 + n] - ----------------------------------------- - 
                                                      8 + 9 n
 
                3 (1 + n) (-4 f[n] - 10 f[1 + n] - 3 n f[1 + n])
>               ------------------------------------------------)) / (8 + 9 (1 + n))) / (8 + 9 (2 + n))
                                    8 + 9 n
 
>         + g[4 + n]) p[4]

In[199]:= Expand[%]

                                                                  4 f[n] p[2]   10 f[1 + n] p[2]
Out[199]= f[n] p[0] + g[n] p[0] + f[1 + n] p[1] + g[1 + n] p[1] - ----------- - ---------------- - 
                                                                    8 + 9 n         8 + 9 n
 
     3 n f[1 + n] p[2]                         52 f[n] p[3]               12 n f[n] p[3]
>    ----------------- + g[2 + n] p[2] + ------------------------- + ------------------------- - 
          8 + 9 n                        (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))
 
     4 f[1 + n] p[3]       130 f[1 + n] p[3]          69 n f[1 + n] p[3]
>    --------------- + ------------------------- + ------------------------- + 
      8 + 9 (1 + n)    (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))
 
           2
        9 n  f[1 + n] p[3]                             16 f[n] p[4]
>    ------------------------- + g[3 + n] p[3] + ------------------------- - 
     (8 + 9 n) (8 + 9 (1 + n))                   (8 + 9 n) (8 + 9 (2 + n))
 
                   832 f[n] p[4]                              348 n f[n] p[4]
>    ----------------------------------------- - ----------------------------------------- - 
     (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                      2
                  36 n  f[n] p[4]                    40 f[1 + n] p[4]
>    ----------------------------------------- + ------------------------- + 
     (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (2 + n))
 
        12 n f[1 + n] p[4]              64 f[1 + n] p[4]                 12 n f[1 + n] p[4]
>    ------------------------- + ------------------------------- + ------------------------------- - 
     (8 + 9 n) (8 + 9 (2 + n))   (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                2080 f[1 + n] p[4]                         1494 n f[1 + n] p[4]
>    ----------------------------------------- - ----------------------------------------- - 
     (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                    2                                           3
               351 n  f[1 + n] p[4]                         27 n  f[1 + n] p[4]
>    ----------------------------------------- - ----------------------------------------- + 
     (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
>    g[4 + n] p[4]

In[200]:= Cases[%, f[_], Infinity]

Out[200]= {f[n], f[1 + n], f[n], f[1 + n], f[1 + n], f[n], f[n], f[1 + n], f[1 + n], f[1 + n], 
 
>    f[1 + n], f[n], f[n], f[n], f[n], f[1 + n], f[1 + n], f[1 + n], f[1 + n], f[1 + n], f[1 + n], 
 
>    f[1 + n], f[1 + n]}

In[201]:= Union[%]

Out[201]= {f[n], f[1 + n]}

In[202]:= Solve[rec2==0, g[n+2]] /. n -> n - 2 + i // First

Out[202]= {g[i + n] -> 
 
      -2 g[-2 + i + n] - 2 (-2 + i + n) g[-2 + i + n] - 3 g[-1 + i + n] - 5 (-2 + i + n) g[-1 + i + n]
>     ------------------------------------------------------------------------------------------------}
                                             2 + 3 (-2 + i + n)

In[203]:= %199 /. First[%202 /. i -> 4] /. First[%202 /. i -> 3] /. First[%202 /. i -> 2]

                                                                  4 f[n] p[2]   10 f[1 + n] p[2]
Out[203]= f[n] p[0] + g[n] p[0] + f[1 + n] p[1] + g[1 + n] p[1] - ----------- - ---------------- - 
                                                                    8 + 9 n         8 + 9 n
 
     3 n f[1 + n] p[2]   (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n]) p[2]
>    ----------------- + ----------------------------------------------------- + 
          8 + 9 n                               2 + 3 n
 
           52 f[n] p[3]               12 n f[n] p[3]         4 f[1 + n] p[3]
>    ------------------------- + ------------------------- - --------------- + 
     (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))    8 + 9 (1 + n)
 
                                                                   2
         130 f[1 + n] p[3]          69 n f[1 + n] p[3]          9 n  f[1 + n] p[3]
>    ------------------------- + ------------------------- + ------------------------- + 
     (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))
 
                                          3 (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n])
>    ((-2 g[1 + n] - 2 (1 + n) g[1 + n] - -------------------------------------------------- - 
                                                               2 + 3 n
 
          5 (1 + n) (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n])
>         ----------------------------------------------------------) p[3]) / (2 + 3 (1 + n)) + 
                                   2 + 3 n
 
           16 f[n] p[4]                        832 f[n] p[4]
>    ------------------------- - ----------------------------------------- - 
     (8 + 9 n) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                                                                  2
                  348 n f[n] p[4]                             36 n  f[n] p[4]
>    ----------------------------------------- - ----------------------------------------- + 
     (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
         40 f[1 + n] p[4]           12 n f[1 + n] p[4]              64 f[1 + n] p[4]
>    ------------------------- + ------------------------- + ------------------------------- + 
     (8 + 9 n) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (2 + n))   (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
           12 n f[1 + n] p[4]                     2080 f[1 + n] p[4]
>    ------------------------------- - ----------------------------------------- - 
     (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                                                                2
               1494 n f[1 + n] p[4]                        351 n  f[1 + n] p[4]
>    ----------------------------------------- - ----------------------------------------- - 
     (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                    3
                27 n  f[1 + n] p[4]
>    ----------------------------------------- + 
     (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
       -2 (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n])
>    ((--------------------------------------------------- - 
                             2 + 3 n
 
          2 (2 + n) (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n])
>         ---------------------------------------------------------- - 
                                   2 + 3 n
 
                                                 3 (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n])
>         (3 (-2 g[1 + n] - 2 (1 + n) g[1 + n] - -------------------------------------------------- - 
                                                                      2 + 3 n
 
               5 (1 + n) (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n])
>              ----------------------------------------------------------)) / (2 + 3 (1 + n)) - 
                                        2 + 3 n
 
>         (5 (2 + n) (-2 g[1 + n] - 2 (1 + n) g[1 + n] - 
 
               3 (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n])
>              -------------------------------------------------- - 
                                    2 + 3 n
 
               5 (1 + n) (-2 g[n] - 2 n g[n] - 3 g[1 + n] - 5 n g[1 + n])
>              ----------------------------------------------------------)) / (2 + 3 (1 + n))) p[4]) / 
                                        2 + 3 n
 
>     (2 + 3 (2 + n))

In[204]:= Union[Cases[%, g[_], Infinity]]

Out[204]= {g[n], g[1 + n]}

In[205]:= Collect[%%, {f[_], g[_]}]

                           3 p[2]    5 n p[2]      2 p[3]       2 (1 + n) p[3]
Out[205]= g[1 + n] (p[1] - ------- - -------- - ------------- - -------------- + 
                           2 + 3 n   2 + 3 n    2 + 3 (1 + n)   2 + 3 (1 + n)
 
                 9 p[3]                     15 n p[3]                15 (1 + n) p[3]
>       ------------------------- + ------------------------- + ------------------------- + 
        (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (1 + n))
 
            25 n (1 + n) p[3]                6 p[4]                     10 n p[4]
>       ------------------------- + ------------------------- + ------------------------- + 
        (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (2 + n))
 
             6 (2 + n) p[4]             10 n (2 + n) p[4]                   6 p[4]
>       ------------------------- + ------------------------- + ------------------------------- + 
        (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                6 (1 + n) p[4]                    10 (2 + n) p[4]
>       ------------------------------- + ------------------------------- + 
        (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
            10 (1 + n) (2 + n) p[4]                        27 p[4]
>       ------------------------------- - ----------------------------------------- - 
        (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                        45 n p[4]                                45 (1 + n) p[4]
>       ----------------------------------------- - ----------------------------------------- - 
        (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                    75 n (1 + n) p[4]                            45 (2 + n) p[4]
>       ----------------------------------------- - ----------------------------------------- - 
        (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                    75 n (2 + n) p[4]                        75 (1 + n) (2 + n) p[4]
>       ----------------------------------------- - ----------------------------------------- - 
        (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
               125 n (1 + n) (2 + n) p[4]
>       -----------------------------------------) + 
        (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                  2 p[2]    2 n p[2]            6 p[3]                     6 n p[3]
>    g[n] (p[0] - ------- - -------- + ------------------------- + ------------------------- + 
                  2 + 3 n   2 + 3 n    (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (1 + n))
 
             10 (1 + n) p[3]            10 n (1 + n) p[3]                4 p[4]
>       ------------------------- + ------------------------- + ------------------------- + 
        (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (2 + n))
 
                4 n p[4]                 4 (2 + n) p[4]             4 n (2 + n) p[4]
>       ------------------------- + ------------------------- + ------------------------- - 
        (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (2 + n))
 
                         18 p[4]                                    18 n p[4]
>       ----------------------------------------- - ----------------------------------------- - 
        (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                     30 (1 + n) p[4]                            30 n (1 + n) p[4]
>       ----------------------------------------- - ----------------------------------------- - 
        (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                     30 (2 + n) p[4]                            30 n (2 + n) p[4]
>       ----------------------------------------- - ----------------------------------------- - 
        (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                 50 (1 + n) (2 + n) p[4]                    50 n (1 + n) (2 + n) p[4]
>       ----------------------------------------- - -----------------------------------------) + 
        (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                  4 p[2]             52 p[3]                    12 n p[3]
>    f[n] (p[0] - ------- + ------------------------- + ------------------------- + 
                  8 + 9 n   (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))
 
                 16 p[4]                            832 p[4]
>       ------------------------- - ----------------------------------------- - 
        (8 + 9 n) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                                                                       2
                       348 n p[4]                                  36 n  p[4]
>       ----------------------------------------- - -----------------------------------------) + 
        (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                      10 p[2]   3 n p[2]      4 p[3]               130 p[3]
>    f[1 + n] (p[1] - ------- - -------- - ------------- + ------------------------- + 
                      8 + 9 n   8 + 9 n    8 + 9 (1 + n)   (8 + 9 n) (8 + 9 (1 + n))
 
                                               2
                69 n p[3]                   9 n  p[3]                    40 p[4]
>       ------------------------- + ------------------------- + ------------------------- + 
        (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (2 + n))
 
                12 n p[4]                       64 p[4]
>       ------------------------- + ------------------------------- + 
        (8 + 9 n) (8 + 9 (2 + n))   (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                   12 n p[4]                              2080 p[4]
>       ------------------------------- - ----------------------------------------- - 
        (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                                                                        2
                       1494 n p[4]                                 351 n  p[4]
>       ----------------------------------------- - ----------------------------------------- - 
        (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                           3
                       27 n  p[4]
>       -----------------------------------------)
        (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))

In[206]:= sys = Thread[(Coefficient[%, #]& /@ {f[n], f[n+1], g[n], g[n+1]})==0]

                  4 p[2]             52 p[3]                    12 n p[3]
Out[206]= {p[0] - ------- + ------------------------- + ------------------------- + 
                  8 + 9 n   (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))
 
                16 p[4]                            832 p[4]
>      ------------------------- - ----------------------------------------- - 
       (8 + 9 n) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                                                                      2
                      348 n p[4]                                  36 n  p[4]
>      ----------------------------------------- - ----------------------------------------- == 0, 
       (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
            10 p[2]   3 n p[2]      4 p[3]               130 p[3]
>    p[1] - ------- - -------- - ------------- + ------------------------- + 
            8 + 9 n   8 + 9 n    8 + 9 (1 + n)   (8 + 9 n) (8 + 9 (1 + n))
 
                                              2
               69 n p[3]                   9 n  p[3]                    40 p[4]
>      ------------------------- + ------------------------- + ------------------------- + 
       (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (1 + n))   (8 + 9 n) (8 + 9 (2 + n))
 
               12 n p[4]                       64 p[4]                          12 n p[4]
>      ------------------------- + ------------------------------- + ------------------------------- - 
       (8 + 9 n) (8 + 9 (2 + n))   (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                       2080 p[4]                                  1494 n p[4]
>      ----------------------------------------- - ----------------------------------------- - 
       (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
                           2                                          3
                      351 n  p[4]                                 27 n  p[4]
>      ----------------------------------------- - ----------------------------------------- == 0, 
       (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))   (8 + 9 n) (8 + 9 (1 + n)) (8 + 9 (2 + n))
 
            2 p[2]    2 n p[2]            6 p[3]                     6 n p[3]
>    p[0] - ------- - -------- + ------------------------- + ------------------------- + 
            2 + 3 n   2 + 3 n    (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (1 + n))
 
            10 (1 + n) p[3]            10 n (1 + n) p[3]                4 p[4]
>      ------------------------- + ------------------------- + ------------------------- + 
       (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (2 + n))
 
               4 n p[4]                 4 (2 + n) p[4]             4 n (2 + n) p[4]
>      ------------------------- + ------------------------- + ------------------------- - 
       (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (2 + n))
 
                        18 p[4]                                    18 n p[4]
>      ----------------------------------------- - ----------------------------------------- - 
       (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                    30 (1 + n) p[4]                            30 n (1 + n) p[4]
>      ----------------------------------------- - ----------------------------------------- - 
       (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                    30 (2 + n) p[4]                            30 n (2 + n) p[4]
>      ----------------------------------------- - ----------------------------------------- - 
       (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                50 (1 + n) (2 + n) p[4]                    50 n (1 + n) (2 + n) p[4]
>      ----------------------------------------- - ----------------------------------------- == 0, 
       (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
            3 p[2]    5 n p[2]      2 p[3]       2 (1 + n) p[3]            9 p[3]
>    p[1] - ------- - -------- - ------------- - -------------- + ------------------------- + 
            2 + 3 n   2 + 3 n    2 + 3 (1 + n)   2 + 3 (1 + n)    (2 + 3 n) (2 + 3 (1 + n))
 
               15 n p[3]                15 (1 + n) p[3]            25 n (1 + n) p[3]
>      ------------------------- + ------------------------- + ------------------------- + 
       (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (1 + n))   (2 + 3 n) (2 + 3 (1 + n))
 
                6 p[4]                     10 n p[4]                6 (2 + n) p[4]
>      ------------------------- + ------------------------- + ------------------------- + 
       (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (2 + n))
 
           10 n (2 + n) p[4]                   6 p[4]                        6 (1 + n) p[4]
>      ------------------------- + ------------------------------- + ------------------------------- + 
       (2 + 3 n) (2 + 3 (2 + n))   (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
               10 (2 + n) p[4]               10 (1 + n) (2 + n) p[4]
>      ------------------------------- + ------------------------------- - 
       (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                        27 p[4]                                    45 n p[4]
>      ----------------------------------------- - ----------------------------------------- - 
       (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                    45 (1 + n) p[4]                            75 n (1 + n) p[4]
>      ----------------------------------------- - ----------------------------------------- - 
       (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                    45 (2 + n) p[4]                            75 n (2 + n) p[4]
>      ----------------------------------------- - ----------------------------------------- - 
       (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))
 
                75 (1 + n) (2 + n) p[4]                   125 n (1 + n) (2 + n) p[4]
>      ----------------------------------------- - ----------------------------------------- == 0}
       (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))   (2 + 3 n) (2 + 3 (1 + n)) (2 + 3 (2 + n))

In[207]:= Solve[%, Table[p[i],{i,0,4}]]

Solve::svars: Equations may not give solutions for all "solve" variables.

                                                2          3         4        5
                    -((-3500 - 19112 n - 25369 n  - 12438 n  - 2475 n  - 162 n ) p[0])
Out[207]= {{p[1] -> ------------------------------------------------------------------, 
                                                               2       3
                                 4 (1 + n) (438 + 887 n + 411 n  + 54 n )
 
                                          2           3           4          5         6
              -((368 - 119156 n - 286440 n  - 259011 n  - 111270 n  - 22797 n  - 1782 n ) p[0])
>     p[2] -> ---------------------------------------------------------------------------------, 
                                                    2         3        4       5
                            8 (876 + 3088 n + 3921 n  + 2228 n  + 573 n  + 54 n )
 
                                         2          3          4         5        6
              -((6684 - 18143 n - 72122 n  - 77045 n  - 37050 n  - 8397 n  - 729 n ) p[0])
>     p[3] -> ----------------------------------------------------------------------------, 
                                                          2        3       4
                          2 (1 + n) (876 + 2212 n + 1709 n  + 519 n  + 54 n )
 
                                          2          3          4         5
              -((19136 - 33416 n - 83358 n  - 54711 n  - 14823 n  - 1458 n ) p[0])
>     p[4] -> --------------------------------------------------------------------}}
                                                  2        3       4
                          8 (876 + 2212 n + 1709 n  + 519 n  + 54 n )

In[208]:= ansatz /. First[%] /. p[0] -> 1

                                           2          3         4        5
                 (-3500 - 19112 n - 25369 n  - 12438 n  - 2475 n  - 162 n ) h[1 + n]
Out[208]= h[n] - ------------------------------------------------------------------- - 
                                                            2       3
                              4 (1 + n) (438 + 887 n + 411 n  + 54 n )
 
                               2           3           4          5         6
     (368 - 119156 n - 286440 n  - 259011 n  - 111270 n  - 22797 n  - 1782 n ) h[2 + n]
>    ---------------------------------------------------------------------------------- - 
                                           2         3        4       5
                   8 (876 + 3088 n + 3921 n  + 2228 n  + 573 n  + 54 n )
 
                              2          3          4         5        6
     (6684 - 18143 n - 72122 n  - 77045 n  - 37050 n  - 8397 n  - 729 n ) h[3 + n]
>    ----------------------------------------------------------------------------- - 
                                                  2        3       4
                  2 (1 + n) (876 + 2212 n + 1709 n  + 519 n  + 54 n )
 
                               2          3          4         5
     (19136 - 33416 n - 83358 n  - 54711 n  - 14823 n  - 1458 n ) h[4 + n]
>    ---------------------------------------------------------------------
                                          2        3       4
                  8 (876 + 2212 n + 1709 n  + 519 n  + 54 n )

In[209]:= Collect[Numerator[Together[%]], h[_], Factor]

                                                2       3
Out[209]= 8 (1 + n) (2 + n) (438 + 887 n + 411 n  + 54 n ) h[n] + 
 
                                        2          3         4        5
>    2 (2 + n) (3500 + 19112 n + 25369 n  + 12438 n  + 2475 n  + 162 n ) h[1 + n] + 
 
                                2           3           4          5         6
>    (-368 + 119156 n + 286440 n  + 259011 n  + 111270 n  + 22797 n  + 1782 n ) h[2 + n] + 
 
                                 2          3          4         5        6
>    4 (-6684 + 18143 n + 72122 n  + 77045 n  + 37050 n  + 8397 n  + 729 n ) h[3 + n] + 
 
                                                      2       3
>    (1 + n) (8 + 3 n) (26 + 9 n) (-92 + 227 n + 249 n  + 54 n ) h[4 + n]

In[210]:= ansatz

Out[210]= h[n] p[0] + h[1 + n] p[1] + h[2 + n] p[2] + h[3 + n] p[3] + h[4 + n] p[4]

In[211]:=