(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 113553, 3484] NotebookOptionsPosition[ 102731, 3277] NotebookOutlinePosition[ 103218, 3295] CellTagsIndexPosition[ 103175, 3292] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{"Exit", "[", "]"}]], "Input", CellChangeTimes->{{3.827398416270541*^9, 3.827398417518792*^9}}, CellLabel->"In[19]:=",ExpressionUUID->"851884f2-edd5-4c2d-9f36-5bd9b7a85eec"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"<<", "Sigma.m"}]], "Input", CellChangeTimes->{{3.828588960235929*^9, 3.828588961922735*^9}, { 3.8316223191519547`*^9, 3.831622319885988*^9}, {3.831695355435175*^9, 3.8316953564021*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"d1b6b1e4-eb24-487b-9e19-61e898f3794f"], Cell[BoxData[ RowBox[{ StyleBox["Sigma", FontColor->RGBColor[1, 0, 0]], RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"-", " ", "A"}], " ", "summation", " ", "package", " ", "by", " ", "Carsten", " ", "Schneider"}], " ", "\[LongDash]", " ", RowBox[{"\[Copyright]", " ", "RISC"}], " ", "\[LongDash]", " ", RowBox[{"V", " ", "2.85", " ", RowBox[{"(", RowBox[{ RowBox[{"June", " ", "2"}], ",", " ", "2021"}], ")"}], " "}]}]}], ButtonBox[ StyleBox["Help", "Hyperlink", FontVariations->{"Underline"->True}], Appearance->{Automatic, None, "Normal", Automatic}, ButtonData:>{"", ""}, ButtonFunction:>Sigma`Version`Private`SigmaHelp[], Evaluator->Automatic]}]], "Print", CellFrame->0.5, CellChangeTimes->{3.831695359479046*^9}, FontColor->RGBColor[0, 0, 0], Background->RGBColor[0.796887, 0.789075, 0.871107], ButtonBoxOptions->{ Active->True},ExpressionUUID->"8bc9d068-7ad0-4b56-b962-d121c81db79f"] }, Open ]], Cell[CellGroupData[{ Cell["The term algebra", "Chapter", CellChangeTimes->{{3.827918391802244*^9, 3.8279184268944263`*^9}, { 3.828588864626494*^9, 3.82858886540082*^9}, 3.828588954775313*^9},ExpressionUUID->"459d8630-4cb8-4f0c-8838-\ 938ac9845f5b"], Cell[CellGroupData[{ Cell["Defining objects in Sigma", "Subsection", CellChangeTimes->{{3.82791899333857*^9, 3.827918996968924*^9}, { 3.828588877794149*^9, 3.828588878159463*^9}, {3.828588958025045*^9, 3.828588958589932*^9}},ExpressionUUID->"f91662f3-4ffb-4cbd-a924-\ 61fa36336114"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"P", "=", RowBox[{ RowBox[{ RowBox[{"SigmaProduct", "[", RowBox[{"k", ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "n"}], "}"}]}], "]"}], "^", "2"}], RowBox[{"SigmaPower", "[", RowBox[{ RowBox[{"-", "1"}], ",", "n"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.8279186462032137`*^9, 3.827918663545827*^9}, 3.8279187966991787`*^9, {3.827918901819124*^9, 3.827918902712126*^9}}, CellLabel->"In[2]:=",ExpressionUUID->"66c7f248-105c-409f-bdb1-b6cf4f480490"], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Product]", "."], RowBox[{"k", "=", "1"}], "n"], "k"}], ")"}], "2"]}]], "Output", CellChangeTimes->{3.8316953667976522`*^9}, CellLabel->"Out[2]=",ExpressionUUID->"123dcdcf-9dbd-4aba-ae36-f9dad4ae9112"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"E1", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"SigmaFactorial", "[", "k", "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "n"}], "}"}]}], "]"}]}], "\n"}]], "Input", CellChangeTimes->{{3.8279184659994087`*^9, 3.827918476668209*^9}}, CellLabel->"In[3]:=",ExpressionUUID->"cdf2189f-2fab-401b-b60e-9bf6d818c2a8"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{"k", SubscriptBox["!", "."]}]}]], "Output", CellChangeTimes->{3.83169536692137*^9}, CellLabel->"Out[3]=",ExpressionUUID->"c5e36c55-d128-45d2-a655-77c3b55125c5"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"GetFullDefinition", "[", "E1", "]"}]], "Input", CellChangeTimes->{{3.827918480157784*^9, 3.827918486774253*^9}}, CellLabel->"In[4]:=",ExpressionUUID->"42a9e20e-5ee9-4507-b7e1-b963b309eaf5"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Product]", "."], RowBox[{ SubscriptBox["\[Omicron]", "1"], "=", "1"}], "k"], SubscriptBox["\[Omicron]", "1"]}]}]], "Output", CellChangeTimes->{3.831695366952636*^9}, CellLabel->"Out[4]=",ExpressionUUID->"0472079d-4317-4721-9d1f-52ddae94e364"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"E1", "/.", RowBox[{"n", "\[Rule]", "5"}]}]], "Input", CellChangeTimes->{{3.8279184906551867`*^9, 3.827918494834299*^9}}, CellLabel->"In[5]:=",ExpressionUUID->"8b877440-fdfa-4e1b-ad06-8d412e7e8cc9"], Cell[BoxData["153"], "Output", CellChangeTimes->{3.831695366977838*^9}, CellLabel->"Out[5]=",ExpressionUUID->"b9aca63a-4c27-4fda-955c-30f9d50ba688"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"E2", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"1", "/", "i"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "k"}], "}"}]}], "]"}], " ", RowBox[{ RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"i", "^", "3"}]}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "k"}], "}"}]}], "]"}], "/", RowBox[{"(", RowBox[{"k", "+", "1"}], ")"}]}]}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.827918506114643*^9, 3.827918529145743*^9}}, CellLabel->"In[6]:=",ExpressionUUID->"81fbce5f-089e-41d5-9823-956c744a20d2"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}]], "Output", CellChangeTimes->{3.831695367008788*^9}, CellLabel->"Out[6]=",ExpressionUUID->"f1dfb6bc-862c-49ba-8309-4b4f463abe1b"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"E2", "/.", RowBox[{"n", "\[Rule]", "4"}]}]], "Input", CellChangeTimes->{{3.8279185334918528`*^9, 3.8279185387826567`*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"65536035-af1f-4c40-bdd4-16bb1fbe157a"], Cell[BoxData[ FractionBox["14417", "6912"]], "Output", CellChangeTimes->{3.831695367044433*^9}, CellLabel->"Out[7]=",ExpressionUUID->"4e848326-e9b8-472d-9665-0a38fe148ec3"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"E3", "=", RowBox[{"E1", "*", RowBox[{"(", RowBox[{"E1", "+", "E2"}], ")"}]}]}]], "Input", CellChangeTimes->{{3.827918567154867*^9, 3.827918597599455*^9}}, CellLabel->"In[8]:=",ExpressionUUID->"6ef87d02-6ba2-4e65-b5ff-680037702a17"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{"k", SubscriptBox["!", "."]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{"k", SubscriptBox["!", "."]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}]}], ")"}]}]], "Output", CellChangeTimes->{3.8316953670795794`*^9}, CellLabel->"Out[8]=",ExpressionUUID->"97777b37-2ebd-4a3b-a9eb-47627b686e9a"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"E3", "/.", RowBox[{"n", "\[Rule]", "4"}]}]], "Input", CellChangeTimes->{{3.82791860025425*^9, 3.827918602903414*^9}}, CellLabel->"In[9]:=",ExpressionUUID->"75005778-1b7e-4f08-8a99-e8b09e75f474"], Cell[BoxData[ FractionBox["2667643", "2304"]], "Output", CellChangeTimes->{3.83169536711758*^9}, CellLabel->"Out[9]=",ExpressionUUID->"a8d52c27-ab66-46c8-b67b-30704d278899"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"E3", "=", RowBox[{"CollectProdSum", "[", "E3", "]"}]}]], "Input", CellChangeTimes->{{3.82791860483445*^9, 3.82791861233134*^9}}, CellLabel->"In[10]:=",ExpressionUUID->"51c5aabc-c553-471c-a146-99226db20810"], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{"k", SubscriptBox["!", "."]}]}], ")"}], "2"], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{"k", SubscriptBox["!", "."]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}], ")"}]}]}]], "Output", CellChangeTimes->{3.83169536716084*^9}, CellLabel->"Out[10]=",ExpressionUUID->"4b9b0f33-f17d-487d-bd07-e71ea0beee7f"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The problem SigmaReduce", "Subsection", CellChangeTimes->{{3.827919006098606*^9, 3.827919009782981*^9}},ExpressionUUID->"1d2860c1-3e4a-40bc-855a-\ 7423b34aa01a"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A1", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"1", "/", "i"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "k"}], "}"}]}], "]"}], RowBox[{ RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"i", "^", "3"}]}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "k"}], "}"}]}], "]"}], "/", RowBox[{"(", RowBox[{"k", "+", "1"}], ")"}]}]}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.8191765609624443`*^9, 3.819176601254879*^9}}, CellLabel->"In[11]:=",ExpressionUUID->"425a78a9-cb98-45fb-a6e6-78a30511735c"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}]], "Output", CellChangeTimes->{3.831695367199843*^9}, CellLabel->"Out[11]=",ExpressionUUID->"368a0dfe-252e-4ae6-a096-0c768fdf5645"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"A2", "=", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "5"]]}], "-", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "4"]]}], RowBox[{"1", "+", "n"}]], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "4"]]}], "j"]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], SuperscriptBox["j", "2"]]}], "+", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], RowBox[{"1", "+", "n"}]], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "4"]]}], "+", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], RowBox[{"1", "+", "n"}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], "k"]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], "k"]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.819176449434576*^9, 3.819176467368972*^9}}, CellLabel->"In[12]:=",ExpressionUUID->"58e7a766-98ca-4a0a-989f-688b3536f48d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"A1", ",", "A2"}], "}"}], "/.", RowBox[{"n", "\[Rule]", "4"}]}]], "Input", CellChangeTimes->{{3.8279182816735363`*^9, 3.82791828982588*^9}}, CellLabel->"In[13]:=",ExpressionUUID->"bfe104f0-6b23-4c4c-99d3-3ace36728362"], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["14417", "6912"], ",", FractionBox["14417", "6912"]}], "}"}]], "Output", CellChangeTimes->{3.8316953672577963`*^9}, CellLabel->"Out[13]=",ExpressionUUID->"9900d1b6-8b38-404b-a6d4-8e6d6fa2a77b"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"B1", ",", "B2"}], "}"}], "=", RowBox[{"SigmaReduce", "[", RowBox[{ RowBox[{"{", RowBox[{"A1", ",", "A2"}], "}"}], ",", "n"}], "]"}]}]], "Input", CellChangeTimes->{{3.8191764272075377`*^9, 3.819176433637816*^9}, { 3.8191764743297367`*^9, 3.819176492127537*^9}}, CellLabel->"In[14]:=",ExpressionUUID->"7a5fd869-602b-493a-bccb-3fa172ee7ac3"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}]]}]}], "}"}]], "Output", CellChangeTimes->{3.8316953696620197`*^9}, CellLabel->"Out[14]=",ExpressionUUID->"16d49442-504a-470a-9b88-d2ac845b79c7"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"W", "=", RowBox[{"Complement", "[", RowBox[{ RowBox[{"Variables", "[", RowBox[{"NShift", "[", RowBox[{ RowBox[{"{", RowBox[{"B1", ",", "B2"}], "}"}], ",", "n"}], "]"}], "]"}], ",", RowBox[{"{", "n", "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.8279190182305326`*^9, 3.827919056649541*^9}}, CellLabel->"In[15]:=",ExpressionUUID->"400fea04-d5f6-42ef-b976-6903a446c250"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}]]}]}], "}"}]], "Output", CellChangeTimes->{3.83169536976283*^9}, CellLabel->"Out[15]=",ExpressionUUID->"07b473cc-8efb-491f-a597-9b22396b4176"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"B2", ",", "B1"}], "}"}], "=", RowBox[{"SigmaReduce", "[", RowBox[{ RowBox[{"{", RowBox[{"A2", ",", "A1"}], "}"}], ",", "n"}], "]"}]}]], "Input", CellChangeTimes->{{3.8191765075745573`*^9, 3.8191765130240917`*^9}, { 3.819177279056683*^9, 3.819177287290031*^9}}, CellLabel->"In[16]:=",ExpressionUUID->"d3d91d94-9cef-4cbc-9ad1-5f13021da5c1"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{"1", "+", "n"}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{"1", "+", "n"}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}]}], "}"}]], "Output", CellChangeTimes->{3.831695371960094*^9}, CellLabel->"Out[16]=",ExpressionUUID->"76b24814-2a55-4261-bf20-3ff1e939738e"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"W", "=", RowBox[{"Complement", "[", RowBox[{ RowBox[{"Variables", "[", RowBox[{"NShift", "[", RowBox[{ RowBox[{"{", RowBox[{"B1", ",", "B2"}], "}"}], ",", "n"}], "]"}], "]"}], ",", RowBox[{"{", "n", "}"}]}], "]"}]}]], "Input", CellLabel->"In[17]:=",ExpressionUUID->"c5ac63ec-7e18-4bd9-b5de-e72811199250"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}], "}"}]], "Output", CellChangeTimes->{3.831695372103735*^9}, CellLabel->"Out[17]=",ExpressionUUID->"2dc6fcd8-d9de-46e9-ab43-9ccfee7cd2c2"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", RowBox[{ RowBox[{"A2", "-", "A1"}], ",", "n"}], "]"}]], "Input", CellChangeTimes->{{3.819176522007575*^9, 3.819176530720066*^9}}, CellLabel->"In[18]:=",ExpressionUUID->"652ed41e-1888-41fe-92f9-a7227b0570e3"], Cell[BoxData["0"], "Output", CellChangeTimes->{3.831695374313547*^9}, CellLabel->"Out[18]=",ExpressionUUID->"b244ca41-0c24-4f6b-b82e-759db9ff15ea"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"varL", "=", RowBox[{"Append", "[", RowBox[{ RowBox[{"Variables", "[", "A2", "]"}], ",", "A1"}], "]"}]}]], "Input", CellChangeTimes->{{3.819177999798284*^9, 3.81917805464017*^9}}, CellLabel->"In[19]:=",ExpressionUUID->"9222af92-c3f1-4086-bccb-ac14201c6226"], Cell[BoxData[ RowBox[{"{", RowBox[{"n", ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "5"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "4"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "4"]]}], "j"]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], SuperscriptBox["j", "2"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "4"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], "k"]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], "k"]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}]}], "}"}]], "Output", CellChangeTimes->{3.831695374362125*^9}, CellLabel->"Out[19]=",ExpressionUUID->"228cbb61-83a6-40b9-84a5-692821224408"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"varLRed", "=", RowBox[{"SigmaReduce", "[", RowBox[{"varL", ",", "n"}], "]"}]}]], "Input", CellChangeTimes->{{3.8191780569386044`*^9, 3.8191780652460938`*^9}, { 3.8191781369521112`*^9, 3.819178141149972*^9}, {3.8191783217720127`*^9, 3.819178322418404*^9}, {3.827918315557498*^9, 3.827918317280294*^9}}, CellLabel->"In[20]:=",ExpressionUUID->"d21be423-8a6b-42a8-9a21-fe498cc0479f"], Cell[BoxData[ RowBox[{"{", RowBox[{"n", ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ",", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}]}], ",", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}]}], ",", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}], ",", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "-", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{"1", "+", "n"}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}]}], "}"}]], "Output", CellChangeTimes->{3.831695376611923*^9}, CellLabel->"Out[20]=",ExpressionUUID->"1fe56d23-7a38-4012-983a-ecd4232ad48e"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"MapThread", "[", RowBox[{"Rule", ",", RowBox[{"{", RowBox[{"varL", ",", "varLRed"}], "}"}]}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.819178072888908*^9, 3.8191781194516487`*^9}, { 3.819178336757028*^9, 3.819178356396052*^9}, {3.827918319762723*^9, 3.827918321048726*^9}}, CellLabel->"In[21]:=",ExpressionUUID->"4bbb293e-5b5d-4db1-bd39-ee5e7a73e766"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ { RowBox[{"n", "\[Rule]", "n"}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "5"]]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "4"]]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "4"]]}], "j"]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], SuperscriptBox["j", "2"]]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "4"]]}], "\[Rule]", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], "\[Rule]", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], "k"]}], "\[Rule]", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], "k"]}], "\[Rule]", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "-", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}], "\[Rule]", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{"1", "+", "n"}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}]}]} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.8316953766882563`*^9}, CellLabel-> "Out[21]//MatrixForm=",ExpressionUUID->"af68bb52-2b2a-4f0a-b353-\ 181325317753"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Symbolic summation in action", "Chapter", CellChangeTimes->{{3.831622247947805*^9, 3.8316222534755993`*^9}},ExpressionUUID->"f908885d-bfa3-4d39-b973-\ 1c685f73134f"], Cell[CellGroupData[{ Cell["Toy examples", "Section", CellChangeTimes->{{3.831622274198222*^9, 3.831622304898262*^9}},ExpressionUUID->"3ec7fa06-35d9-4269-aa2e-\ f49b20fefaa1"], Cell[CellGroupData[{ Cell["Telescoping", "Subsection", CellChangeTimes->{{3.83162427645835*^9, 3.831624277985805*^9}},ExpressionUUID->"6d9045c2-d4c1-4b53-8e37-\ 0ff603bd2631"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"S", "[", RowBox[{"1", ",", "k"}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.831622287298766*^9, 3.8316223137350388`*^9}}, CellLabel->"In[22]:=",ExpressionUUID->"e2c8ef8e-002a-4f41-ab1b-0ba2d5a47ad7"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{"S", "[", RowBox[{"1", ",", "k"}], "]"}]}]], "Output", CellChangeTimes->{3.831636315810925*^9, 3.831695381778981*^9}, CellLabel->"Out[22]=",ExpressionUUID->"e79340c8-ffcf-4267-b19c-88f4c47a4253"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", "mySum", "]"}]], "Input", CellChangeTimes->{{3.831624224425425*^9, 3.831624228276979*^9}}, CellLabel->"In[23]:=",ExpressionUUID->"2db5acb2-23dc-4674-9bbe-d5df57fbfd67"], Cell[BoxData[ RowBox[{ RowBox[{"-", "n"}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}]}]}]], "Output", CellChangeTimes->{3.831636323103984*^9, 3.831695382002977*^9}, CellLabel->"Out[23]=",ExpressionUUID->"62df2090-dead-4cc6-b9a4-ef6c6e4f9e78"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{"SigmaFactorial", "[", "k", "]"}], " ", "k"}], ",", RowBox[{"{", RowBox[{"k", ",", "a", ",", "b"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.831624211999302*^9}}, CellLabel->"In[24]:=",ExpressionUUID->"bc1178b4-b258-40cf-9be3-6bd7bf5f4bc2"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "a"}], "b"], RowBox[{"k", " ", RowBox[{"k", SubscriptBox["!", "."]}]}]}]], "Output", CellChangeTimes->{3.831636349065555*^9, 3.831695382034697*^9}, CellLabel->"Out[24]=",ExpressionUUID->"f7e2dcfb-f4f8-4bee-9c21-7a6dce0678d4"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", "mySum", "]"}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.8316242126586523`*^9}}, CellLabel->"In[25]:=",ExpressionUUID->"d88ff7ab-bb59-49a8-ba05-b944dce7359e"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "a"}], " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "a"}], ")"}], SubscriptBox["!", "."]}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "b"}], ")"}], " ", RowBox[{"b", SubscriptBox["!", "."]}]}]}]], "Output", CellChangeTimes->{3.83163635454921*^9, 3.83169538217455*^9}, CellLabel->"Out[25]=",ExpressionUUID->"10f60be6-a0ba-462b-a201-e4fe4a7ae019"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{"SigmaPower", "[", RowBox[{ RowBox[{"-", "1"}], ",", "k"}], "]"}], " ", RowBox[{"SigmaBinomial", "[", RowBox[{"n", ",", "k"}], "]"}], " ", RowBox[{"SigmaHNumber", "[", "k", "]"}]}], ",", RowBox[{"{", RowBox[{"k", ",", "a", ",", "b"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.831624213292328*^9}}, CellLabel->"In[26]:=",ExpressionUUID->"38e23792-f224-4bba-b914-3a3f637e23f4"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "a"}], "b"], RowBox[{ SubscriptBox["H", "k"], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["k", "."]]}]}]], "Output", CellChangeTimes->{3.831636377164926*^9, 3.831695382207223*^9}, CellLabel->"Out[26]=",ExpressionUUID->"f7db102d-e3dc-4619-adfe-19f700ec29b4"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", "mySum", "]"}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.831624213913733*^9}}, CellLabel->"In[27]:=",ExpressionUUID->"245256e3-7de1-4f5e-8f29-50c5266baf76"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "a", "-", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"a", "-", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { RowBox[{ RowBox[{"-", "1"}], "+", "a"}]} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["a", "."]]}], RowBox[{"a", " ", SuperscriptBox["n", "2"]}]]}], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "a", "+", "n"}], ")"}], " ", SubscriptBox["H", "a"], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { RowBox[{ RowBox[{"-", "1"}], "+", "a"}]} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["a", "."]]}], "n"], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "b"}], "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"b"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["b", "."]]}], SuperscriptBox["n", "2"]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "b"}], "+", "n"}], ")"}], " ", SubscriptBox["H", "b"], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"b"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["b", "."]]}], "n"]}]], "Output", CellChangeTimes->{3.831636388778604*^9, 3.831695382906118*^9}, CellLabel->"Out[27]=",ExpressionUUID->"9eca07ce-0c18-4801-b190-afcad63a17c7"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"SigmaBinomial", "[", RowBox[{"n", ",", "k"}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "0", ",", "r"}], "}"}]}], "]"}], "^", "2"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "b"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.8316242145103703`*^9}}, CellLabel->"In[28]:=",ExpressionUUID->"af75c852-4958-4d3e-b6b3-691592942265"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"r", "=", "0"}], "b"], SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "r"], SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], "."]}], ")"}], "2"]}]], "Output", CellChangeTimes->{3.83163640452861*^9, 3.831695383001484*^9}, CellLabel->"Out[28]=",ExpressionUUID->"101a6230-bc42-458a-9a69-b079b252321c"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"res", "=", RowBox[{"SigmaReduce", "[", "mySum", "]"}]}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.8316242151100273`*^9}}, CellLabel->"In[29]:=",ExpressionUUID->"0d224b46-7d22-46ed-9dc4-6daea32b5c33"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "b"}], "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"b"} }], ")"}], "."], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "b"], SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."]}], ")"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"2", "+", RowBox[{"2", " ", "b"}], "-", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "b"], SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."]}], ")"}], "2"]}], "-", RowBox[{ FractionBox["1", "2"], " ", "n", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "b"], SuperscriptBox[ RowBox[{"(", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."], ")"}], "2"]}], ")"}]}]}]], "Output", CellChangeTimes->{3.8316364204483967`*^9, 3.831695383357109*^9}, CellLabel->"Out[29]=",ExpressionUUID->"42faf624-bc57-4fe2-8927-410a089ebe62"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"mySum", "-", "res"}], "}"}], "/.", RowBox[{"{", RowBox[{"b", "\[Rule]", "10"}], "}"}]}], "//", "Together"}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.8316242162011957`*^9}, { 3.831636446746657*^9, 3.8316364570174*^9}}, CellLabel->"In[30]:=",ExpressionUUID->"22cd9332-4904-4fc0-abd8-7924473c4ca0"], Cell[BoxData[ RowBox[{"{", "0", "}"}]], "Output", CellChangeTimes->{{3.8316364509481153`*^9, 3.831636457740253*^9}, 3.831695383440839*^9}, CellLabel->"Out[30]=",ExpressionUUID->"a305e6a9-8e53-4e96-bb12-56bfcfd75bc9"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Definite summation", "Subsection", CellChangeTimes->{{3.831624203440839*^9, 3.831624216849856*^9}, 3.8316242828476543`*^9},ExpressionUUID->"75de7937-3466-4af4-9301-\ 40848fbc0599"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"SigmaHNumber", "[", "k", "]"}], "^", "2"}], " ", RowBox[{"SigmaBinomial", "[", RowBox[{"n", ",", "k"}], "]"}]}], ",", RowBox[{"{", RowBox[{"k", ",", "0", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.8316242174324083`*^9}, { 3.831695406843995*^9, 3.831695408795796*^9}}, CellLabel->"In[36]:=",ExpressionUUID->"0eac2d65-2dfc-445a-bf38-a5bc1a8bb50a"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "n"], RowBox[{ SubsuperscriptBox["H", "k", "2"], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], "."]}]}]], "Output", CellChangeTimes->{{3.831695383473031*^9, 3.831695409236664*^9}}, CellLabel->"Out[36]=",ExpressionUUID->"2cb7a3ec-2951-4b4d-b5e3-315735cccfc4"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", "mySum", "]"}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.831624218146708*^9}}, CellLabel->"In[37]:=",ExpressionUUID->"6396d67d-db28-48e3-a9b2-8b8bde677f01"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "n"], RowBox[{ SubsuperscriptBox["H", SubscriptBox["\[Iota]", "1"], "2"], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."]}]}]], "Output", CellChangeTimes->{{3.831695383656279*^9, 3.831695410404294*^9}}, CellLabel->"Out[37]=",ExpressionUUID->"aef5e8a9-cf36-4fa7-a223-01df451f5c7b"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"rec", "=", RowBox[{"GenerateRecurrence", "[", "mySum", "]"}]}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.8316242187834053`*^9}}, CellLabel->"In[38]:=",ExpressionUUID->"82e79f20-00e6-47d1-8ba2-576f3eb0ecca"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "8"}], " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"3", "+", "n"}], ")"}], " ", RowBox[{"SUM", "[", "n", "]"}]}], "+", RowBox[{"4", " ", RowBox[{"(", RowBox[{"29", "+", RowBox[{"25", " ", "n"}], "+", RowBox[{"5", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}], "-", RowBox[{"2", " ", RowBox[{"(", RowBox[{"8", "+", RowBox[{"3", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{"10", "+", RowBox[{"3", " ", "n"}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"2", "+", "n"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"86", "+", RowBox[{"49", " ", "n"}], "+", RowBox[{"7", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"3", "+", "n"}], "]"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"4", "+", "n"}], ")"}], "2"], " ", RowBox[{"SUM", "[", RowBox[{"4", "+", "n"}], "]"}]}]}], "\[Equal]", RowBox[{"-", "1"}]}], "}"}]], "Output", CellChangeTimes->{{3.8316953839386806`*^9, 3.8316954132960663`*^9}}, CellLabel->"Out[38]=",ExpressionUUID->"0347e85a-4b7d-41d7-a55d-23c8023d20da"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"recSol", "=", RowBox[{"SolveRecurrence", "[", RowBox[{ RowBox[{"rec", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"SUM", "[", "n", "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.8316242194162197`*^9}}, CellLabel->"In[39]:=",ExpressionUUID->"58ed9239-ce3a-48ae-b128-4c828c74c378"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", SuperscriptBox["2", UnderscriptBox["n", "."]]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", RowBox[{ SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], "-", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], "2"], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubscriptBox["\[Iota]", "2"]]}], RowBox[{ SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{ SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", RowBox[{ SubsuperscriptBox["\[Iota]", "1", "2"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}], ")"}]}], "-", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", RowBox[{ SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}], ")"}]}], "+", RowBox[{"3", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], RowBox[{ SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}], ")"}]}]}], ")"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.831695384055179*^9, 3.8316954172457447`*^9}, CellLabel->"Out[39]=",ExpressionUUID->"a9895de0-301e-43ee-b238-d618bf8da42d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindLinearCombination", "[", RowBox[{"recSol", ",", "mySum", ",", "n", ",", "4"}], "]"}]], "Input", CellChangeTimes->{{3.831624203440839*^9, 3.8316242194162197`*^9}, { 3.831695425235219*^9, 3.831695426001748*^9}}, CellLabel->"In[41]:=",ExpressionUUID->"d77910d2-4451-4f0f-9711-b71b675ffa0c"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], "+", RowBox[{ SuperscriptBox["2", UnderscriptBox["n", "."]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], "2"]}], "-", RowBox[{"2", " ", SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", RowBox[{ SubsuperscriptBox["\[Iota]", "1", "2"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}], ")"}]}], "-", RowBox[{"2", " ", SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", RowBox[{ SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}], ")"}]}], "+", RowBox[{"3", " ", SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], RowBox[{ SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}], ")"}]}], "-", RowBox[{ SuperscriptBox["2", UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubscriptBox["\[Iota]", "2"]]}], RowBox[{ SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox["2", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]]}], ")"}]}]}]], "Output", CellChangeTimes->{ 3.8316953841067057`*^9, {3.831695417413629*^9, 3.831695426566051*^9}}, CellLabel->"Out[41]=",ExpressionUUID->"591abeb5-30a3-413c-8fda-378079103556"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The full machinery", "Section", CellChangeTimes->{{3.83162225869281*^9, 3.831622284289261*^9}, { 3.831624308295051*^9, 3.831624314736467*^9}},ExpressionUUID->"a71d7b28-b50d-4f6f-92ec-\ 0af2b04673e7"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"<<", "EvaluateMultiSums.m"}]], "Input", CellChangeTimes->{{3.831622591328903*^9, 3.831622598042297*^9}}, CellLabel->"In[13]:=",ExpressionUUID->"6b497308-8917-453b-a098-027cb4be0136"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ "EvaluateMultiSums", " ", "by", " ", "Carsten", " ", "Schneider", " ", RowBox[{"(", RowBox[{ RowBox[{"still", " ", "under", " ", "development"}], "..."}], ")"}]}], " ", "\[LongDash]", " ", RowBox[{"\[Copyright]", " ", "RISC"}], " ", "\[LongDash]", " ", RowBox[{"V", " ", "0.993", " ", RowBox[{"(", RowBox[{ RowBox[{"September", " ", "18"}], ",", " ", "2020"}], ")"}]}]}]], "Print",\ CellFrame->0.5, CellChangeTimes->{3.8316371618108807`*^9}, FontColor->RGBColor[0, 0, 0], Background->RGBColor[0.796887, 0.789075, 0.871107], ButtonBoxOptions->{ Active->True},ExpressionUUID->"e324f51b-b3aa-4ddc-b35e-c0953a78dc03"], Cell[BoxData["\<\"A note for the usage of EvaluateMultiSums: If more \ complicated nested sums arise, one should also activate the package \ HarmonicSums by Jakob Ablinger (for speed ups and appropriate limit \ calculations for infinite sums).\"\>"], "Print", CellChangeTimes->{3.831637161965884*^9}, CellLabel-> "During evaluation of \ In[13]:=",ExpressionUUID->"94976cb0-0963-417a-bb79-8248d2cbb95d"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The warm-up example", "Subsubsection", CellChangeTimes->{{3.831624321622698*^9, 3.831624327922365*^9}},ExpressionUUID->"0b5cfb3b-1194-4072-b005-\ 1c316ad5cfa4"], Cell[BoxData[ RowBox[{ RowBox[{"summand", "=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"j", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"N", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"j", "+", "k", "+", "N"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"2", "+", RowBox[{"2", " ", "j"}], "+", "k", "+", "N"}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "j", "+", "N"}], ")"}]}]], "-", RowBox[{"S", "[", RowBox[{"1", ",", "j"}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k"}]}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "N"}]}], "]"}], "-", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k", "+", "N"}]}], "]"}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "N"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "N"}], ")"}], SubscriptBox["!", "."]}]}], ")"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.695358996200685*^9, 3.695358999573455*^9}}, CellLabel->"In[14]:=",ExpressionUUID->"93f01bcf-cfc4-4b2e-8154-edbf65c117ae"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "\[Infinity]"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "0"}], "\[Infinity]"], "summand"}]}]}]], "Input", CellChangeTimes->{{3.695359054949247*^9, 3.695359058352847*^9}}, CellLabel->"In[15]:=",ExpressionUUID->"b5ec48f5-fa1f-4495-9c52-ff1ae788211e"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "\[Infinity]"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "0"}], "\[Infinity]"], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"j", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"N", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"j", "+", "k", "+", "N"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"2", "+", RowBox[{"2", " ", "j"}], "+", "k", "+", "N"}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "j", "+", "N"}], ")"}]}]], "-", RowBox[{"S", "[", RowBox[{"1", ",", "j"}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k"}]}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "N"}]}], "]"}], "-", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k", "+", "N"}]}], "]"}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "N"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "N"}], ")"}], SubscriptBox["!", "."]}]}], ")"}]}]}]}]], "Output", CellChangeTimes->{3.8316371689161167`*^9}, CellLabel->"Out[15]=",ExpressionUUID->"8ed28314-4cf2-4294-9b1d-f13506fab261"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EvaluateMultiSum", "[", RowBox[{"mySum", ",", RowBox[{"{", "}"}], ",", RowBox[{"{", "N", "}"}], ",", RowBox[{"{", "1", "}"}], ",", RowBox[{"{", "Infinity", "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.695359067258122*^9, 3.695359084209755*^9}}, CellLabel->"In[16]:=",ExpressionUUID->"f1f079bf-5668-4a45-968b-62c5e0aef345"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"3", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "N"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], RowBox[{"2", " ", "N", " ", RowBox[{"(", RowBox[{"1", "+", "N"}], ")"}]}]], "+", FractionBox[ SuperscriptBox[ RowBox[{"S", "[", RowBox[{"1", ",", "N"}], "]"}], "2"], RowBox[{"2", " ", "N", " ", RowBox[{"(", RowBox[{"1", "+", "N"}], ")"}]}]]}]], "Output", CellChangeTimes->{3.83163721235248*^9}, CellLabel->"Out[16]=",ExpressionUUID->"a3463eac-322b-4f37-9eae-9f8e2910c8f7"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The triple sum", "Subsection", CellChangeTimes->{{3.8316229865058203`*^9, 3.831622988671309*^9}},ExpressionUUID->"3c14e639-a713-486d-b2dc-\ 63d98f3625a4"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"summand", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "^", "r"}], " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "j"}], "+", "n", "-", "2"}], ")"}], "!"}], " ", RowBox[{"Binomial", "[", RowBox[{ RowBox[{"j", "+", "1"}], ",", "r"}], "]"}], " ", RowBox[{ RowBox[{"r", "!"}], "/", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "j"}], "+", "n", "+", "r"}], ")"}], "!"}]}], " ", RowBox[{ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "^", "s"}], " ", RowBox[{ RowBox[{"Binomial", "[", RowBox[{ RowBox[{ RowBox[{"-", "j"}], "+", "n", "+", "r", "-", "2"}], ",", "s"}], "]"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"n", "-", "s"}], ")"}], " ", RowBox[{"(", RowBox[{"s", "+", "1"}], ")"}]}], ")"}]}]}]}]], "Input", CellChangeTimes->{{3.831622989459194*^9, 3.831622999190834*^9}}, CellLabel->"In[17]:=",ExpressionUUID->"39198eab-81af-4ded-b667-adc8ce2ef7ae"], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], RowBox[{"r", "+", "s"}]], " ", RowBox[{"Binomial", "[", RowBox[{ RowBox[{"1", "+", "j"}], ",", "r"}], "]"}], " ", RowBox[{"Binomial", "[", RowBox[{ RowBox[{ RowBox[{"-", "2"}], "-", "j", "+", "n", "+", "r"}], ",", "s"}], "]"}], " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "-", "j", "+", "n"}], ")"}], "!"}], " ", RowBox[{"r", "!"}]}], RowBox[{ RowBox[{"(", RowBox[{"n", "-", "s"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "s"}], ")"}], " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "j"}], "+", "n", "+", "r"}], ")"}], "!"}]}]]], "Output", CellChangeTimes->{3.8316372305046787`*^9}, CellLabel->"Out[17]=",ExpressionUUID->"563d48ee-474f-465a-91ef-dd132bc74270"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"res", "=", RowBox[{ RowBox[{"EvaluateMultiSum", "[", RowBox[{"summand", ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"s", ",", "0", ",", RowBox[{ RowBox[{"-", "j"}], "+", "n", "+", "r", "-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", RowBox[{"j", "+", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "0", ",", RowBox[{"n", "-", "2"}]}], "}"}]}], "}"}], ",", RowBox[{"{", "n", "}"}], ",", RowBox[{"{", "2", "}"}]}], "]"}], "//", "Timing"}]}]], "Input", CellChangeTimes->{{3.831622989459194*^9, 3.831623007427988*^9}}, CellLabel->"In[18]:=",ExpressionUUID->"446f258d-7c61-4e12-9dcf-f2a7e1a6f651"], Cell[BoxData[ RowBox[{"{", RowBox[{"75.601439`", ",", RowBox[{ FractionBox[ RowBox[{ RowBox[{"-", "1"}], "-", "n", "-", SuperscriptBox["n", "2"]}], RowBox[{ SuperscriptBox["n", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n", "+", SuperscriptBox["n", "2"]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]]}], RowBox[{ SuperscriptBox["n", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "-", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], RowBox[{"1", "+", "n"}]], "+", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "2"]], "-", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], RowBox[{"1", "+", "n"}]]}]}], "}"}]], "Output", CellChangeTimes->{3.831637308313405*^9}, CellLabel->"Out[18]=",ExpressionUUID->"d7715fd7-59b7-477b-921a-7949271bfd8d"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{1114.5, 600.}, WindowMargins->{{Automatic, 0}, {0, Automatic}}, TaggingRules->{ "WelcomeScreenSettings" -> {"FEStarting" -> False}, "TryRealOnly" -> False}, FrontEndVersion->"12.1 for Linux x86 (64-bit) (March 18, 2020)", StyleDefinitions->"Default.nb", ExpressionUUID->"e223920c-93f8-4e5a-abe0-78f32606f264" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[558, 20, 197, 3, 29, "Input",ExpressionUUID->"851884f2-edd5-4c2d-9f36-5bd9b7a85eec"], Cell[CellGroupData[{ Cell[780, 27, 293, 5, 29, "Input",ExpressionUUID->"d1b6b1e4-eb24-487b-9e19-61e898f3794f"], Cell[1076, 34, 972, 26, 40, "Print",ExpressionUUID->"8bc9d068-7ad0-4b56-b962-d121c81db79f"] }, Open ]], Cell[CellGroupData[{ Cell[2085, 65, 235, 4, 69, "Chapter",ExpressionUUID->"459d8630-4cb8-4f0c-8838-938ac9845f5b"], Cell[CellGroupData[{ Cell[2345, 73, 267, 4, 54, "Subsection",ExpressionUUID->"f91662f3-4ffb-4cbd-a924-61fa36336114"], Cell[CellGroupData[{ Cell[2637, 81, 527, 13, 29, "Input",ExpressionUUID->"66c7f248-105c-409f-bdb1-b6cf4f480490"], Cell[3167, 96, 433, 13, 58, "Output",ExpressionUUID->"123dcdcf-9dbd-4aba-ae36-f9dad4ae9112"] }, Open ]], Cell[CellGroupData[{ Cell[3637, 114, 383, 9, 51, "Input",ExpressionUUID->"cdf2189f-2fab-401b-b60e-9bf6d818c2a8"], Cell[4023, 125, 292, 8, 58, "Output",ExpressionUUID->"c5e36c55-d128-45d2-a655-77c3b55125c5"] }, Open ]], Cell[CellGroupData[{ Cell[4352, 138, 215, 3, 29, "Input",ExpressionUUID->"42a9e20e-5ee9-4507-b7e1-b963b309eaf5"], Cell[4570, 143, 430, 12, 61, "Output",ExpressionUUID->"0472079d-4317-4721-9d1f-52ddae94e364"] }, Open ]], Cell[CellGroupData[{ Cell[5037, 160, 226, 4, 29, "Input",ExpressionUUID->"8b877440-fdfa-4e1b-ad06-8d412e7e8cc9"], Cell[5266, 166, 150, 2, 33, "Output",ExpressionUUID->"b9aca63a-4c27-4fda-955c-30f9d50ba688"] }, Open ]], Cell[CellGroupData[{ Cell[5453, 173, 775, 22, 32, "Input",ExpressionUUID->"81fbce5f-089e-41d5-9823-956c744a20d2"], Cell[6231, 197, 690, 22, 76, "Output",ExpressionUUID->"f1dfb6bc-862c-49ba-8309-4b4f463abe1b"] }, Open ]], Cell[CellGroupData[{ Cell[6958, 224, 228, 4, 29, "Input",ExpressionUUID->"65536035-af1f-4c40-bdd4-16bb1fbe157a"], Cell[7189, 230, 175, 3, 48, "Output",ExpressionUUID->"4e848326-e9b8-472d-9665-0a38fe148ec3"] }, Open ]], Cell[CellGroupData[{ Cell[7401, 238, 269, 6, 29, "Input",ExpressionUUID->"6ef87d02-6ba2-4e65-b5ff-680037702a17"], Cell[7673, 246, 1164, 38, 89, "Output",ExpressionUUID->"97777b37-2ebd-4a3b-a9eb-47627b686e9a"] }, Open ]], Cell[CellGroupData[{ Cell[8874, 289, 223, 4, 29, "Input",ExpressionUUID->"75005778-1b7e-4f08-8a99-e8b09e75f474"], Cell[9100, 295, 176, 3, 49, "Output",ExpressionUUID->"a8d52c27-ab66-46c8-b67b-30704d278899"] }, Open ]], Cell[CellGroupData[{ Cell[9313, 303, 235, 4, 29, "Input",ExpressionUUID->"51c5aabc-c553-471c-a146-99226db20810"], Cell[9551, 309, 1217, 40, 89, "Output",ExpressionUUID->"4b9b0f33-f17d-487d-bd07-e71ea0beee7f"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[10817, 355, 171, 3, 54, "Subsection",ExpressionUUID->"1d2860c1-3e4a-40bc-855a-7423b34aa01a"], Cell[CellGroupData[{ Cell[11013, 362, 773, 22, 32, "Input",ExpressionUUID->"425a78a9-cb98-45fb-a6e6-78a30511735c"], Cell[11789, 386, 691, 22, 76, "Output",ExpressionUUID->"368a0dfe-252e-4ae6-a096-0c768fdf5645"] }, Open ]], Cell[12495, 411, 3404, 111, 90, "Input",ExpressionUUID->"58e7a766-98ca-4a0a-989f-688b3536f48d"], Cell[CellGroupData[{ Cell[15924, 526, 274, 6, 29, "Input",ExpressionUUID->"bfe104f0-6b23-4c4c-99d3-3ace36728362"], Cell[16201, 534, 254, 6, 48, "Output",ExpressionUUID->"9900d1b6-8b38-404b-a6d4-8e6d6fa2a77b"] }, Open ]], Cell[CellGroupData[{ Cell[16492, 545, 411, 10, 29, "Input",ExpressionUUID->"7a5fd869-602b-493a-bccb-3fa172ee7ac3"], Cell[16906, 557, 1870, 58, 79, "Output",ExpressionUUID->"16d49442-504a-470a-9b88-d2ac845b79c7"] }, Open ]], Cell[CellGroupData[{ Cell[18813, 620, 442, 11, 29, "Input",ExpressionUUID->"400fea04-d5f6-42ef-b976-6903a446c250"], Cell[19258, 633, 1446, 45, 79, "Output",ExpressionUUID->"07b473cc-8efb-491f-a597-9b22396b4176"] }, Open ]], Cell[CellGroupData[{ Cell[20741, 683, 411, 10, 29, "Input",ExpressionUUID->"d3d91d94-9cef-4cbc-9ad1-5f13021da5c1"], Cell[21155, 695, 6156, 192, 154, "Output",ExpressionUUID->"76b24814-2a55-4261-bf20-3ff1e939738e"] }, Open ]], Cell[CellGroupData[{ Cell[27348, 892, 374, 10, 29, "Input",ExpressionUUID->"c5ac63ec-7e18-4bd9-b5de-e72811199250"], Cell[27725, 904, 2574, 81, 79, "Output",ExpressionUUID->"2dc6fcd8-d9de-46e9-ab43-9ccfee7cd2c2"] }, Open ]], Cell[CellGroupData[{ Cell[30336, 990, 258, 5, 29, "Input",ExpressionUUID->"652ed41e-1888-41fe-92f9-a7227b0570e3"], Cell[30597, 997, 149, 2, 33, "Output",ExpressionUUID->"b244ca41-0c24-4f6b-b82e-759db9ff15ea"] }, Open ]], Cell[CellGroupData[{ Cell[30783, 1004, 293, 6, 29, "Input",ExpressionUUID->"9222af92-c3f1-4086-bccb-ac14201c6226"], Cell[31079, 1012, 3688, 124, 94, "Output",ExpressionUUID->"228cbb61-83a6-40b9-84a5-692821224408"] }, Open ]], Cell[CellGroupData[{ Cell[34804, 1141, 417, 7, 29, "Input",ExpressionUUID->"d21be423-8a6b-42a8-9a21-fe498cc0479f"], Cell[35224, 1150, 12399, 388, 244, "Output",ExpressionUUID->"1fe56d23-7a38-4012-983a-ecd4232ad48e"] }, Open ]], Cell[CellGroupData[{ Cell[47660, 1543, 430, 10, 29, "Input",ExpressionUUID->"4bbb293e-5b5d-4db1-bd39-ee5e7a73e766"], Cell[48093, 1555, 19192, 545, 618, "Output",ExpressionUUID->"af68bb52-2b2a-4f0a-b353-181325317753"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[67346, 2107, 175, 3, 69, "Chapter",ExpressionUUID->"f908885d-bfa3-4d39-b973-1c685f73134f"], Cell[CellGroupData[{ Cell[67546, 2114, 157, 3, 67, "Section",ExpressionUUID->"3ec7fa06-35d9-4269-aa2e-f49b20fefaa1"], Cell[CellGroupData[{ Cell[67728, 2121, 158, 3, 54, "Subsection",ExpressionUUID->"6d9045c2-d4c1-4b53-8e37-0ff603bd2631"], Cell[CellGroupData[{ Cell[67911, 2128, 376, 9, 29, "Input",ExpressionUUID->"e2c8ef8e-002a-4f41-ab1b-0ba2d5a47ad7"], Cell[68290, 2139, 327, 8, 58, "Output",ExpressionUUID->"e79340c8-ffcf-4267-b19c-88f4c47a4253"] }, Open ]], Cell[CellGroupData[{ Cell[68654, 2152, 213, 3, 29, "Input",ExpressionUUID->"2db5acb2-23dc-4674-9bbe-d5df57fbfd67"], Cell[68870, 2157, 337, 9, 33, "Output",ExpressionUUID->"62df2090-dead-4cc6-b9a4-ef6c6e4f9e78"] }, Open ]], Cell[CellGroupData[{ Cell[69244, 2171, 387, 9, 29, "Input",ExpressionUUID->"bc1178b4-b258-40cf-9be3-6bd7bf5f4bc2"], Cell[69634, 2182, 341, 9, 60, "Output",ExpressionUUID->"f7e2dcfb-f4f8-4bee-9c21-7a6dce0678d4"] }, Open ]], Cell[CellGroupData[{ Cell[70012, 2196, 215, 3, 29, "Input",ExpressionUUID->"d88ff7ab-bb59-49a8-ba05-b944dce7359e"], Cell[70230, 2201, 464, 15, 33, "Output",ExpressionUUID->"10f60be6-a0ba-462b-a201-e4fe4a7ae019"] }, Open ]], Cell[CellGroupData[{ Cell[70731, 2221, 555, 14, 29, "Input",ExpressionUUID->"38e23792-f224-4bba-b914-3a3f637e23f4"], Cell[71289, 2237, 520, 17, 60, "Output",ExpressionUUID->"f7db102d-e3dc-4619-adfe-19f700ec29b4"] }, Open ]], Cell[CellGroupData[{ Cell[71846, 2259, 213, 3, 29, "Input",ExpressionUUID->"245256e3-7de1-4f5e-8f29-50c5266baf76"], Cell[72062, 2264, 1869, 70, 67, "Output",ExpressionUUID->"9eca07ce-0c18-4801-b190-afcad63a17c7"] }, Open ]], Cell[CellGroupData[{ Cell[73968, 2339, 551, 14, 29, "Input",ExpressionUUID->"af75c852-4958-4d3e-b6b3-691592942265"], Cell[74522, 2355, 540, 17, 60, "Output",ExpressionUUID->"101a6230-bc42-458a-9a69-b079b252321c"] }, Open ]], Cell[CellGroupData[{ Cell[75099, 2377, 240, 4, 29, "Input",ExpressionUUID->"0d224b46-7d22-46ed-9dc4-6daea32b5c33"], Cell[75342, 2383, 1631, 58, 61, "Output",ExpressionUUID->"42faf624-bc57-4fe2-8927-410a089ebe62"] }, Open ]], Cell[CellGroupData[{ Cell[77010, 2446, 386, 9, 29, "Input",ExpressionUUID->"22cd9332-4904-4fc0-abd8-7924473c4ca0"], Cell[77399, 2457, 223, 4, 33, "Output",ExpressionUUID->"a305e6a9-8e53-4e96-bb12-56bfcfd75bc9"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[77671, 2467, 191, 3, 54, "Subsection",ExpressionUUID->"75de7937-3466-4af4-9301-40848fbc0599"], Cell[CellGroupData[{ Cell[77887, 2474, 533, 13, 29, "Input",ExpressionUUID->"0eac2d65-2dfc-445a-bf38-a5bc1a8bb50a"], Cell[78423, 2489, 425, 13, 58, "Output",ExpressionUUID->"2cb7a3ec-2951-4b4d-b5e3-315735cccfc4"] }, Open ]], Cell[CellGroupData[{ Cell[78885, 2507, 213, 3, 29, "Input",ExpressionUUID->"6396d67d-db28-48e3-a9b2-8b8bde677f01"], Cell[79101, 2512, 519, 16, 59, "Output",ExpressionUUID->"aef5e8a9-cf36-4fa7-a223-01df451f5c7b"] }, Open ]], Cell[CellGroupData[{ Cell[79657, 2533, 247, 4, 29, "Input",ExpressionUUID->"82e79f20-00e6-47d1-8ba2-576f3eb0ecca"], Cell[79907, 2539, 1424, 44, 36, "Output",ExpressionUUID->"0347e85a-4b7d-41d7-a55d-23c8023d20da"] }, Open ]], Cell[CellGroupData[{ Cell[81368, 2588, 357, 8, 29, "Input",ExpressionUUID->"58ed9239-ce3a-48ae-b128-4c828c74c378"], Cell[81728, 2598, 5021, 150, 146, "Output",ExpressionUUID->"a9895de0-301e-43ee-b238-d618bf8da42d"] }, Open ]], Cell[CellGroupData[{ Cell[86786, 2753, 322, 5, 29, "Input",ExpressionUUID->"d77910d2-4451-4f0f-9711-b71b675ffa0c"], Cell[87111, 2760, 3493, 114, 112, "Output",ExpressionUUID->"591abeb5-30a3-413c-8fda-378079103556"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[90665, 2881, 211, 4, 67, "Section",ExpressionUUID->"a71d7b28-b50d-4f6f-92ec-0af2b04673e7"], Cell[CellGroupData[{ Cell[90901, 2889, 208, 3, 29, "Input",ExpressionUUID->"6b497308-8917-453b-a098-027cb4be0136"], Cell[CellGroupData[{ Cell[91134, 2896, 695, 19, 40, "Print",ExpressionUUID->"e324f51b-b3aa-4ddc-b35e-c0953a78dc03"], Cell[91832, 2917, 406, 7, 43, "Print",ExpressionUUID->"94976cb0-0963-417a-bb79-8248d2cbb95d"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[92287, 2930, 170, 3, 45, "Subsubsection",ExpressionUUID->"0b5cfb3b-1194-4072-b005-1c316ad5cfa4"], Cell[92460, 2935, 1723, 52, 72, "Input",ExpressionUUID->"93f01bcf-cfc4-4b2e-8154-edbf65c117ae"], Cell[CellGroupData[{ Cell[94208, 2991, 444, 11, 55, "Input",ExpressionUUID->"b5ec48f5-fa1f-4495-9c52-ff1ae788211e"], Cell[94655, 3004, 1888, 58, 82, "Output",ExpressionUUID->"8ed28314-4cf2-4294-9b1d-f13506fab261"] }, Open ]], Cell[CellGroupData[{ Cell[96580, 3067, 369, 8, 29, "Input",ExpressionUUID->"f1f079bf-5668-4a45-968b-62c5e0aef345"], Cell[96952, 3077, 734, 23, 72, "Output",ExpressionUUID->"a3463eac-322b-4f37-9eae-9f8e2910c8f7"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[97735, 3106, 164, 3, 54, "Subsection",ExpressionUUID->"3c14e639-a713-486d-b2dc-63d98f3625a4"], Cell[CellGroupData[{ Cell[97924, 3113, 1065, 35, 31, "Input",ExpressionUUID->"39198eab-81af-4ded-b667-adc8ce2ef7ae"], Cell[98992, 3150, 894, 30, 52, "Output",ExpressionUUID->"563d48ee-474f-465a-91ef-dd132bc74270"] }, Open ]], Cell[CellGroupData[{ Cell[99923, 3185, 772, 20, 29, "Input",ExpressionUUID->"446f258d-7c61-4e12-9dcf-f2a7e1a6f651"], Cell[100698, 3207, 1981, 64, 73, "Output",ExpressionUUID->"d7715fd7-59b7-477b-921a-7949271bfd8d"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] } ] *)