(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 68644, 1684]*) (*NotebookOutlinePosition[ 69281, 1706]*) (* CellTagsIndexPosition[ 69237, 1702]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Summing the geometric series and generating functions", "Subtitle", CellAutoOverwrite->False], Cell["\<\ Ivan Cnop Vrije universiteit Brussel icnop@vub.ac.be\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["A geometric series", "Section"], Cell["We know the sum of a geometric series and its partial sums", "Text"], Cell[BoxData[ \(\[Sum]\+\(j = 0\)\%\[Infinity] x\^j\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(\(\[Sum]\+\(j = 0\)\%n\)\(x\^j\)\(\ \ \ \)\)\)], "Input", Evaluatable->False], Cell["For |x| small enough these are ", "Text"], Cell[BoxData[ \(1\/\(1 - x\)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(1 - \ x\^\(n + 1\)\)\/\(1 - x\)\)], "Input", Evaluatable->False], Cell["\<\ and they are the generating function for the sequence \ 1,1,1,1,1,1,1,1,1,1,1,.... We next try to sum the series that is the generating function for \ succesive k th powers of natural numbers j :\ \>", "Text"], Cell[BoxData[ StyleBox[\(\[Sum]\+\(j = 0\)\%\[Infinity]\( j\^k\) x\^j\), FontColor->RGBColor[1, 0, 0]]], "Input", Evaluatable->False], Cell[TextData[{ "We suspect it may be another fraction with a power of 1-x in the \ denominator.\n", StyleBox["The numerator will be called a Euler polynomial, and its \ coefficients are called Euler coefficients.", FontWeight->"Bold"] }], "Text"], Cell["Mathematica knows these:", "Text"], Cell[CellGroupData[{ Cell["Table[Sum[j^k*x^j, {j, 0, Infinity}], {k, 0, 12}]//TableForm", "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(1\/\(1 - x\)\)}, {\(x\/\((\(-1\) + x)\)\^2\)}, {\(\(\(-x\) - x\^2\)\/\((\(-1\) + x)\)\^3\)}, {\(\(x + 4\ x\^2 + x\^3\)\/\((\(-1\) + x)\)\^4\)}, {\(\(\(-x\) - 11\ x\^2 - 11\ x\^3 - x\^4\)\/\((\(-1\) + x)\)\^5\)}, {\(\(x + 26\ x\^2 + 66\ x\^3 + 26\ x\^4 + x\^5\)\/\((\(-1\) + x)\)\^6\)}, {\(\(\(-x\) - 57\ x\^2 - 302\ x\^3 - 302\ x\^4 - 57\ x\^5 - x\^6\)\/\((\(-1\) + x)\)\^7\)}, {\(\(x + 120\ x\^2 + 1191\ x\^3 + 2416\ x\^4 + 1191\ x\^5 + 120\ x\^6 + x\^7\)\/\((\(-1\) + x)\)\^8\)}, {\(\(\(-x\) - 247\ x\^2 - 4293\ x\^3 - 15619\ x\^4 - 15619\ x\^5 - 4293\ x\^6 - 247\ x\^7 - x\^8\)\/\((\(-1\) + x)\)\^9\)}, {\(\(x + 502\ x\^2 + 14608\ x\^3 + 88234\ x\^4 + 156190\ x\^5 + 88234\ x\^6 + 14608\ x\^7 + 502\ x\^8 + x\^9\)\/\((\(-1\) + x)\)\^10\)}, {\(\(\(-x\) - 1013\ x\^2 - 47840\ x\^3 - 455192\ x\^4 - 1310354\ x\^5 - 1310354\ x\^6 - 455192\ x\^7 - 47840\ x\^8 - 1013\ x\^9 - x\^10\)\/\((\(-1\) + x)\)\^11\)}, {\(\(x + 2036\ x\^2 + 152637\ x\^3 + 2203488\ x\^4 + 9738114\ x\^5 + 15724248\ x\^6 + 9738114\ x\^7 + 2203488\ x\^8 + 152637\ x\^9 + 2036\ x\^10 + x\^11\)\/\((\(-1\) + x)\)\^12\)}, {\(\(x + 4083\ x\^2 + 478271\ x\^3 + 10187685\ x\^4 + 66318474\ x\^5 + 162512286\ x\^6 + 162512286\ x\^7 + 66318474\ x\^8 + 10187685\ x\^9 + 478271\ x\^10 + 4083\ x\^11 + x\^12\)\/\((1 - x)\)\^13\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Power[ Plus[ 1, Times[ -1, x]], -1], Times[ Power[ Plus[ -1, x], -2], x], Times[ Power[ Plus[ -1, x], -3], Plus[ Times[ -1, x], Times[ -1, Power[ x, 2]]]], Times[ Power[ Plus[ -1, x], -4], Plus[ x, Times[ 4, Power[ x, 2]], Power[ x, 3]]], Times[ Power[ Plus[ -1, x], -5], Plus[ Times[ -1, x], Times[ -11, Power[ x, 2]], Times[ -11, Power[ x, 3]], Times[ -1, Power[ x, 4]]]], Times[ Power[ Plus[ -1, x], -6], Plus[ x, Times[ 26, Power[ x, 2]], Times[ 66, Power[ x, 3]], Times[ 26, Power[ x, 4]], Power[ x, 5]]], Times[ Power[ Plus[ -1, x], -7], Plus[ Times[ -1, x], Times[ -57, Power[ x, 2]], Times[ -302, Power[ x, 3]], Times[ -302, Power[ x, 4]], Times[ -57, Power[ x, 5]], Times[ -1, Power[ x, 6]]]], Times[ Power[ Plus[ -1, x], -8], Plus[ x, Times[ 120, Power[ x, 2]], Times[ 1191, Power[ x, 3]], Times[ 2416, Power[ x, 4]], Times[ 1191, Power[ x, 5]], Times[ 120, Power[ x, 6]], Power[ x, 7]]], Times[ Power[ Plus[ -1, x], -9], Plus[ Times[ -1, x], Times[ -247, Power[ x, 2]], Times[ -4293, Power[ x, 3]], Times[ -15619, Power[ x, 4]], Times[ -15619, Power[ x, 5]], Times[ -4293, Power[ x, 6]], Times[ -247, Power[ x, 7]], Times[ -1, Power[ x, 8]]]], Times[ Power[ Plus[ -1, x], -10], Plus[ x, Times[ 502, Power[ x, 2]], Times[ 14608, Power[ x, 3]], Times[ 88234, Power[ x, 4]], Times[ 156190, Power[ x, 5]], Times[ 88234, Power[ x, 6]], Times[ 14608, Power[ x, 7]], Times[ 502, Power[ x, 8]], Power[ x, 9]]], Times[ Power[ Plus[ -1, x], -11], Plus[ Times[ -1, x], Times[ -1013, Power[ x, 2]], Times[ -47840, Power[ x, 3]], Times[ -455192, Power[ x, 4]], Times[ -1310354, Power[ x, 5]], Times[ -1310354, Power[ x, 6]], Times[ -455192, Power[ x, 7]], Times[ -47840, Power[ x, 8]], Times[ -1013, Power[ x, 9]], Times[ -1, Power[ x, 10]]]], Times[ Power[ Plus[ -1, x], -12], Plus[ x, Times[ 2036, Power[ x, 2]], Times[ 152637, Power[ x, 3]], Times[ 2203488, Power[ x, 4]], Times[ 9738114, Power[ x, 5]], Times[ 15724248, Power[ x, 6]], Times[ 9738114, Power[ x, 7]], Times[ 2203488, Power[ x, 8]], Times[ 152637, Power[ x, 9]], Times[ 2036, Power[ x, 10]], Power[ x, 11]]], Times[ Power[ Plus[ 1, Times[ -1, x]], -13], Plus[ x, Times[ 4083, Power[ x, 2]], Times[ 478271, Power[ x, 3]], Times[ 10187685, Power[ x, 4]], Times[ 66318474, Power[ x, 5]], Times[ 162512286, Power[ x, 6]], Times[ 162512286, Power[ x, 7]], Times[ 66318474, Power[ x, 8]], Times[ 10187685, Power[ x, 9]], Times[ 478271, Power[ x, 10]], Times[ 4083, Power[ x, 11]], Power[ x, 12]]]}]]], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Recursive definition of Euler polynomials", "Section", CellAutoOverwrite->False], Cell["\<\ These functies are defined by successive derivatives and \ multiplication by x .\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(Clear[f, l]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(f[0, x_] := 1/\((1 - x)\)\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(f[j_, x_] := \(f[j, x] = x\ D[f[j - 1, x], x]\)\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(l[j_] := CoefficientList[ Series[\((f[j, x]\ \((1 - x)\)^\((j + 1)\))\), {x, 0, j}], x]\)], "Input", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(l[4]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({0, 1, 11, 11, 1}\)], "Output"] }, Closed]], Cell["But this is slow due to the Series computation:", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[l[10]]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({0.8799999999999999`\ Second, {0, 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1}}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Last[%] // TableForm\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[GridBox[{ {"0"}, {"1"}, {"1013"}, {"47840"}, {"455192"}, {"1310354"}, {"1310354"}, {"455192"}, {"47840"}, {"1013"}, {"1"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {0, 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1}]]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(MemoryInUse[]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(3771660\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[l[12]]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({5.5`\ Second, {0, 1, 4083, 478271, 10187685, 66318474, 162512286, 162512286, 66318474, 10187685, 478271, 4083, 1}}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(MemoryInUse[]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(3800520\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Better definition of Euler polynomials", "Section", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell["Working with poynomials", "Subsection"], Cell["\<\ If we know these functions are a polynomial divided by a power of \ (1-x) , we can speed up the definition:\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(Clear[f, l]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(f[0, x_] := 1/\((1 - x)\)\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(f[j_, x_] := \(f[j, x] = Together[x\ D[f[j - 1, x], x]]\)\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(l[j_] := CoefficientList[Together[\((f[j, x]\ \((1 - x)\)^\((j + 1)\))\)], x]\)], "Input", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(l[4]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({0, 1, 11, 11, 1}\)], "Output"] }, Closed]], Cell["This is fast and memory efficient:", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[l[10]]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({0.019999999999999574`\ Second, {0, 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1}}\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(MemoryInUse[]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(3818420\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(First[Timing[l[45]]]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(0.3899999999999988`\ Second\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["A \"half circle\" of Euler coefficients:", "Subsection", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(Last[Timing[l[45]]] // TableForm\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[GridBox[{ {"0"}, {"1"}, {"35184372088786"}, {"2954311088069717613406"}, {"1237804140937294761660912586"}, {"28364767246309997123544180777751"}, {"102639520123344490994216622024361492"}, {"102254802774844556321638487129340075412"}, {"38740977857702396518016707296441283975252"}, {"6833559879257653667981373297500211348721987"}, {"641986842449785445256313534577890326715086242"}, {"35280063431806455060695308677560818666781286278"}, {"1213771252697178941392658775245136125693906246858"}, {"27499980614651715920593300976097910730367835933493"}, {"426469178869633461861116374598942272662024940453808"}, {"4664610845665929059799873899941533119963810570828528"}, {"36840664802320571157446453424745727349198161303280944"}, {"214034884556631242927734754340936689362203252250089434"}, {"928213599155612523056306236775945746648465070018361764"}, {"3039469233842776712049388396035038655486499032335987484"}, {"7581584591564436785502067079250188936883562812951567444"}, {"14499974022298570847733144100206872395904209086984530070"}, {"21358488620335186064121163804266122900510145189843470520"}, {"24294769972958473891093960848611277683172702839268700920"}, {"21358488620335186064121163804266122900510145189843470520"}, {"14499974022298570847733144100206872395904209086984530070"}, {"7581584591564436785502067079250188936883562812951567444"}, {"3039469233842776712049388396035038655486499032335987484"}, {"928213599155612523056306236775945746648465070018361764"}, {"214034884556631242927734754340936689362203252250089434"}, {"36840664802320571157446453424745727349198161303280944"}, {"4664610845665929059799873899941533119963810570828528"}, {"426469178869633461861116374598942272662024940453808"}, {"27499980614651715920593300976097910730367835933493"}, {"1213771252697178941392658775245136125693906246858"}, {"35280063431806455060695308677560818666781286278"}, {"641986842449785445256313534577890326715086242"}, {"6833559879257653667981373297500211348721987"}, {"38740977857702396518016707296441283975252"}, {"102254802774844556321638487129340075412"}, {"102639520123344490994216622024361492"}, {"28364767246309997123544180777751"}, {"1237804140937294761660912586"}, {"2954311088069717613406"}, {"35184372088786"}, {"1"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {0, 1, 35184372088786, 2954311088069717613406, 1237804140937294761660912586, 28364767246309997123544180777751, 102639520123344490994216622024361492, 102254802774844556321638487129340075412, 38740977857702396518016707296441283975252, 6833559879257653667981373297500211348721987, 641986842449785445256313534577890326715086242, 35280063431806455060695308677560818666781286278, 1213771252697178941392658775245136125693906246858, 27499980614651715920593300976097910730367835933493, 426469178869633461861116374598942272662024940453808, 4664610845665929059799873899941533119963810570828528, 36840664802320571157446453424745727349198161303280944, 214034884556631242927734754340936689362203252250089434, 928213599155612523056306236775945746648465070018361764, 3039469233842776712049388396035038655486499032335987484, 7581584591564436785502067079250188936883562812951567444, 14499974022298570847733144100206872395904209086984530070, 21358488620335186064121163804266122900510145189843470520, 24294769972958473891093960848611277683172702839268700920, 21358488620335186064121163804266122900510145189843470520, 14499974022298570847733144100206872395904209086984530070, 7581584591564436785502067079250188936883562812951567444, 3039469233842776712049388396035038655486499032335987484, 928213599155612523056306236775945746648465070018361764, 214034884556631242927734754340936689362203252250089434, 36840664802320571157446453424745727349198161303280944, 4664610845665929059799873899941533119963810570828528, 426469178869633461861116374598942272662024940453808, 27499980614651715920593300976097910730367835933493, 1213771252697178941392658775245136125693906246858, 35280063431806455060695308677560818666781286278, 641986842449785445256313534577890326715086242, 6833559879257653667981373297500211348721987, 38740977857702396518016707296441283975252, 102254802774844556321638487129340075412, 102639520123344490994216622024361492, 28364767246309997123544180777751, 1237804140937294761660912586, 2954311088069717613406, 35184372088786, 1}]]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(MemoryInUse[]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(4049312\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["More Euler coefficients:", "Subsection", CellAutoOverwrite->False], Cell["\<\ After reducing Font Size we can write coefficienten of degree 100 :\ \ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(Last[Timing[l[100]]] // TableForm\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[GridBox[{ {"0"}, {"1"}, {"1267650600228229401496703205275"}, {"515377520732011203003750506714451721535083784075"}, {"1606938044206937145948028954308115559568433722798231162561425"}, { "78886088899093755865619243027863479807547976736296853433607667554\ 79224"}, { "65331782675056474791189022332543454184193492190167780995769976161\ 2311631852456"}, { "32344105244836219596289968744828696579088764165630728396702235487\ 20277070241422750600"}, { "20367092975062717197499859649643254849715739148636494923684379721\ 76471388474497937125232600"}, { "26540826457626248433269837748243195063629116117580136893814757276\ 3100907375497359328820538770500"}, { "99731832736161942945728688300737467829635974446029797343503599604\ 42566628593954317781079447309908844"}, { "13679746440940768562588094769993899893993093825810991130291761145\ 0714961411386431312095036567719572361996"}, { "81431178250067057244927280398828688401420677576959815030264015677\ 2228214306753737182642757381264520835451300"}, { "23963832122062195109613740053134856057066891202000695831354634463\ 75786275983889564736797434093413137302987297000"}, { "38539326257734807064678887540672399165924146363264750274775215566\ 87152958678213588039313018209582087340158031478200"}, { "36638761683944618909497891003353208423450445419339162599306983864\ 50258582031719079944326775858869023957119637510836504"}, { "21919192178217958780366522161168979390494836744850581930424817263\ 69136187949904220388091210344135696891856845240898552456"}, { "86795023606253234830695837823899097311482203138560789614836926539\ 6561673653789388362086147831183169956746595666830985395250"}, { "23708626887311550699574921569477586158445359593391415088829628942\ 7661445144076007836724336676029159920342032633970952598390150"}, { "46230233058739886203383074962229907211471003516169620610365026339\ 070595391022410144087253402634530693944178694912872915408190150"}, { "66220840809739801533001618448668683464067011308583760473488324984\ 90197172507009452337815250315950214208576857197709790453711306994"}, { "71387616874726665714111054153899700355794983292626074718542560699\ 3226220440282861801332577837896896040137322489558074166991304308776"}, { "59124505465532816734151085800412451678935636106766101183880539011\ 210538335101021378668641675343335571208341611366995861223686581726200"}, { "38293133742547939453936865728723340371824235213471858157108481473\ 16010174976564187250083011477404277144933543825476992954269276683051800"}, { "19694079354549032669876228058201322505520271257646322095361594900\ 6376652354493619501386648099948903032752793954480150261445037100626798600"}, { "81508245461361486168380637158887423694921161400831247709437345876\ 04463215938059632640592410303951676068053801339915785064711989279853247124"}, { "27465375710584340975427585836255084225897979864547370575630630643\ 7136406137582198338154436129694200720001956860133256892841199518152685889372"}\ , { "76128014328941790602520709368997903907776210824332322957094472285\ 94990379248165429760777552405729188621973277839721881390779769352798204394300"\ }, { "17515230927803724200373924814142253827208869423370498007686680336\ 816471889734708565576547515589572960730335456488375157108360377620911409721490\ 0"}, { "33720325592135546500248566240392813427150486092943142486969591765\ 308182439495869950739202676572900191294250341585053675998812696504298632194846\ 00"}, { "54711642033190190972334905611816888788163996116333546677841919655\ 367026524232737584032137929760642880732891813207132422147203982304910314713964\ 328"}, { "75292741546080359451393110165393140616132252501963416587980631679\ 711605937224170395696580522896498476869829885149188808262628855262755060047389\ 3832"}, { "88387861006163598775724921605801371553676625389943800888356201900\ 630797179966531386259091876045526356110610156125722877464025327867702734716675\ 70200"}, { "88965486141751036383175237909540923981745517089267900968241339989\ 111095109853202982534247969769503559560900196948791178475874387014255030870719\ 440575"}, { "77132088333660587652235893707350689193252865340675499455854948519\ 074208397760150235932339125279433445530620739811127196466852885029925576499572\ 8801125"}, { "57839588526119005000948467788321728277610713021958218924071103296\ 314336699251078000486320662243608241917385735914199947837288323741417158774226\ 64239893"}, { "37653186594396314358430187770167124233082550387266532535098287163\ 511821352406855069059216088079640980885627834406147907066298898202212097127226\ 290143887"}, { "21350618352576141207111641167016872870072159968924190037934072466\ 624933721828950793617198312005966318224438455012306353088351467665386569169536\ 5310375600"}, { "10576654916974665745717749516079505314767531604505376002813694607\ 673912309612174723739361412259495386285992993331747454003203304402051160429091\ 12797578000"}, { "45896200036662428197828622813661437673427180749125555518533941160\ 635729532677558770916132057714704928913251731773457263144549601964031207017568\ 83646632400"}, { "17487698571834809439529677994034201158360804587530314446523180315\ 521572579782652074113833195334563019008992611212858721736270272356164978858990\ 592751829488"}, { "58632506868535284197864012527661680555969225968971476720760709101\ 013104954161595328151949665795567598109421553092637857062614656841668919340699\ 466745799432"}, { "17330460069601900369470804778970401541878053393158113161184730688\ 922064834793469109307355950203706765190808219866401834760691669175807725129043\ 3643756455000"}, { "45233916295740721235987395758648920422136197018724204721573781206\ 849610694653708031130611245710763696707746915988230100788909191748488982126677\ 3979058139800"}, { "10440568121672226932134901952890892492187897926297333301597837773\ 517731325578465201690081777000852400962161593263415481925549725825192456207305\ 82202330457800"}, { "21336595681791330561985314770797380545431408889953938414850626545\ 670879475679556935677787185749206002834296539307728339457364725304430737373018\ 14087147117968"}, { "38647242126191317230582558403248983949172648333128276320025292412\ 870039683311328255413606839836715078111743844685552445364559788524249027353634\ 62308764842672"}, { "62097662651387570316922063804666429043378055952644638285944521943\ 337942400896198093754515064889633972787410575207507329337991378805028877079349\ 55565305868400"}, { "88570450039352168944505864720787197756640551049760766115737679316\ 246523047802016142846325752497280107012447475799998601735366788736138051930703\ 01329982683600"}, { "11219603743512400677675882526090393853676492681988732262429627696\ 833795295137737461153948617725643423012972548639856537715036982089769132516931\ 815081824648700"}, { "12626583048565731758704803903613273608195677189975104100350973856\ 449564133356757386852425614117017819889941988926638004335803974817249421258019\ 632694790596628"}, { "12626583048565731758704803903613273608195677189975104100350973856\ 449564133356757386852425614117017819889941988926638004335803974817249421258019\ 632694790596628"}, { "11219603743512400677675882526090393853676492681988732262429627696\ 833795295137737461153948617725643423012972548639856537715036982089769132516931\ 815081824648700"}, { "88570450039352168944505864720787197756640551049760766115737679316\ 246523047802016142846325752497280107012447475799998601735366788736138051930703\ 01329982683600"}, { "62097662651387570316922063804666429043378055952644638285944521943\ 337942400896198093754515064889633972787410575207507329337991378805028877079349\ 55565305868400"}, { "38647242126191317230582558403248983949172648333128276320025292412\ 870039683311328255413606839836715078111743844685552445364559788524249027353634\ 62308764842672"}, { "21336595681791330561985314770797380545431408889953938414850626545\ 670879475679556935677787185749206002834296539307728339457364725304430737373018\ 14087147117968"}, { "10440568121672226932134901952890892492187897926297333301597837773\ 517731325578465201690081777000852400962161593263415481925549725825192456207305\ 82202330457800"}, { "45233916295740721235987395758648920422136197018724204721573781206\ 849610694653708031130611245710763696707746915988230100788909191748488982126677\ 3979058139800"}, { "17330460069601900369470804778970401541878053393158113161184730688\ 922064834793469109307355950203706765190808219866401834760691669175807725129043\ 3643756455000"}, { "58632506868535284197864012527661680555969225968971476720760709101\ 013104954161595328151949665795567598109421553092637857062614656841668919340699\ 466745799432"}, { "17487698571834809439529677994034201158360804587530314446523180315\ 521572579782652074113833195334563019008992611212858721736270272356164978858990\ 592751829488"}, { "45896200036662428197828622813661437673427180749125555518533941160\ 635729532677558770916132057714704928913251731773457263144549601964031207017568\ 83646632400"}, { "10576654916974665745717749516079505314767531604505376002813694607\ 673912309612174723739361412259495386285992993331747454003203304402051160429091\ 12797578000"}, { "21350618352576141207111641167016872870072159968924190037934072466\ 624933721828950793617198312005966318224438455012306353088351467665386569169536\ 5310375600"}, { "37653186594396314358430187770167124233082550387266532535098287163\ 511821352406855069059216088079640980885627834406147907066298898202212097127226\ 290143887"}, { "57839588526119005000948467788321728277610713021958218924071103296\ 314336699251078000486320662243608241917385735914199947837288323741417158774226\ 64239893"}, { "77132088333660587652235893707350689193252865340675499455854948519\ 074208397760150235932339125279433445530620739811127196466852885029925576499572\ 8801125"}, { "88965486141751036383175237909540923981745517089267900968241339989\ 111095109853202982534247969769503559560900196948791178475874387014255030870719\ 440575"}, { "88387861006163598775724921605801371553676625389943800888356201900\ 630797179966531386259091876045526356110610156125722877464025327867702734716675\ 70200"}, { "75292741546080359451393110165393140616132252501963416587980631679\ 711605937224170395696580522896498476869829885149188808262628855262755060047389\ 3832"}, { "54711642033190190972334905611816888788163996116333546677841919655\ 367026524232737584032137929760642880732891813207132422147203982304910314713964\ 328"}, { "33720325592135546500248566240392813427150486092943142486969591765\ 308182439495869950739202676572900191294250341585053675998812696504298632194846\ 00"}, { "17515230927803724200373924814142253827208869423370498007686680336\ 816471889734708565576547515589572960730335456488375157108360377620911409721490\ 0"}, { "76128014328941790602520709368997903907776210824332322957094472285\ 94990379248165429760777552405729188621973277839721881390779769352798204394300"\ }, { "27465375710584340975427585836255084225897979864547370575630630643\ 7136406137582198338154436129694200720001956860133256892841199518152685889372"}\ , { "81508245461361486168380637158887423694921161400831247709437345876\ 04463215938059632640592410303951676068053801339915785064711989279853247124"}, { "19694079354549032669876228058201322505520271257646322095361594900\ 6376652354493619501386648099948903032752793954480150261445037100626798600"}, { "38293133742547939453936865728723340371824235213471858157108481473\ 16010174976564187250083011477404277144933543825476992954269276683051800"}, { "59124505465532816734151085800412451678935636106766101183880539011\ 210538335101021378668641675343335571208341611366995861223686581726200"}, { "71387616874726665714111054153899700355794983292626074718542560699\ 3226220440282861801332577837896896040137322489558074166991304308776"}, { "66220840809739801533001618448668683464067011308583760473488324984\ 90197172507009452337815250315950214208576857197709790453711306994"}, { "46230233058739886203383074962229907211471003516169620610365026339\ 070595391022410144087253402634530693944178694912872915408190150"}, { "23708626887311550699574921569477586158445359593391415088829628942\ 7661445144076007836724336676029159920342032633970952598390150"}, { "86795023606253234830695837823899097311482203138560789614836926539\ 6561673653789388362086147831183169956746595666830985395250"}, { "21919192178217958780366522161168979390494836744850581930424817263\ 69136187949904220388091210344135696891856845240898552456"}, { "36638761683944618909497891003353208423450445419339162599306983864\ 50258582031719079944326775858869023957119637510836504"}, { "38539326257734807064678887540672399165924146363264750274775215566\ 87152958678213588039313018209582087340158031478200"}, { "23963832122062195109613740053134856057066891202000695831354634463\ 75786275983889564736797434093413137302987297000"}, { "81431178250067057244927280398828688401420677576959815030264015677\ 2228214306753737182642757381264520835451300"}, { "13679746440940768562588094769993899893993093825810991130291761145\ 0714961411386431312095036567719572361996"}, { "99731832736161942945728688300737467829635974446029797343503599604\ 42566628593954317781079447309908844"}, { "26540826457626248433269837748243195063629116117580136893814757276\ 3100907375497359328820538770500"}, { "20367092975062717197499859649643254849715739148636494923684379721\ 76471388474497937125232600"}, { "32344105244836219596289968744828696579088764165630728396702235487\ 20277070241422750600"}, { "65331782675056474791189022332543454184193492190167780995769976161\ 2311631852456"}, { "78886088899093755865619243027863479807547976736296853433607667554\ 79224"}, {"1606938044206937145948028954308115559568433722798231162561425"}, {"515377520732011203003750506714451721535083784075"}, {"1267650600228229401496703205275"}, {"1"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {0, 1, 1267650600228229401496703205275, 515377520732011203003750506714451721535083784075, 1606938044206937145948028954308115559568433722798231162561425, 7888608889909375586561924302786347980754797673629685343360766755479224\ , 6533178267505647479118902233254345418419349219016778099576997616123116318524\ 56, 32344105244836219596289968744828696579088764165630728396702235487202770702\ 41422750600, 2036709297506271719749985964964325484971573914863649492368437972176471\ 388474497937125232600, 2654082645762624843326983774824319506362911611758013689381475727631009\ 07375497359328820538770500, 9973183273616194294572868830073746782963597444602979734350359960442566\ 628593954317781079447309908844, 1367974644094076856258809476999389989399309382581099113029176114507149\ 61411386431312095036567719572361996, 8143117825006705724492728039882868840142067757695981503026401567722282\ 14306753737182642757381264520835451300, 2396383212206219510961374005313485605706689120200069583135463446375786\ 275983889564736797434093413137302987297000, 3853932625773480706467888754067239916592414636326475027477521556687152\ 958678213588039313018209582087340158031478200, 3663876168394461890949789100335320842345044541933916259930698386450258\ 582031719079944326775858869023957119637510836504, 2191919217821795878036652216116897939049483674485058193042481726369136\ 187949904220388091210344135696891856845240898552456, 8679502360625323483069583782389909731148220313856078961483692653965616\ 73653789388362086147831183169956746595666830985395250, 2370862688731155069957492156947758615844535959339141508882962894276614\ 45144076007836724336676029159920342032633970952598390150, 4623023305873988620338307496222990721147100351616962061036502633907059\ 5391022410144087253402634530693944178694912872915408190150, 6622084080973980153300161844866868346406701130858376047348832498490197\ 172507009452337815250315950214208576857197709790453711306994, 7138761687472666571411105415389970035579498329262607471854256069932262\ 20440282861801332577837896896040137322489558074166991304308776, 5912450546553281673415108580041245167893563610676610118388053901121053\ 8335101021378668641675343335571208341611366995861223686581726200, 3829313374254793945393686572872334037182423521347185815710848147316010\ 174976564187250083011477404277144933543825476992954269276683051800, 1969407935454903266987622805820132250552027125764632209536159490063766\ 52354493619501386648099948903032752793954480150261445037100626798600, 8150824546136148616838063715888742369492116140083124770943734587604463\ 215938059632640592410303951676068053801339915785064711989279853247124, 2746537571058434097542758583625508422589797986454737057563063064371364\ 06137582198338154436129694200720001956860133256892841199518152685889372, 7612801432894179060252070936899790390777621082433232295709447228594990\ 379248165429760777552405729188621973277839721881390779769352798204394300, 1751523092780372420037392481414225382720886942337049800768668033681647\ 18897347085655765475155895729607303354564883751571083603776209114097214900, 3372032559213554650024856624039281342715048609294314248696959176530818\ 243949586995073920267657290019129425034158505367599881269650429863219484600, 5471164203319019097233490561181688878816399611633354667784191965536702\ 6524232737584032137929760642880732891813207132422147203982304910314713964328, 7529274154608035945139311016539314061613225250196341658798063167971160\ 59372241703956965805228964984768698298851491888082626288552627550600473893832, 883878610061635987757249216058013715536766253899438008883562019006307\ 971799665313862590918760455263561106101561257228774640253278677027347166757020\ 0, 889654861417510363831752379095409239817455170892679009682413399891110951098\ 53202982534247969769503559560900196948791178475874387014255030870719440575, 7713208833366058765223589370735068919325286534067549945585494851907420\ 839776015023593233912527943344553062073981112719646685288502992557649957288011\ 25, 57839588526119005000948467788321728277610713021958218924071103296314336699\ 25107800048632066224360824191738573591419994783728832374141715877422664239893, 376531865943963143584301877701671242330825503872665325350982871635118\ 213524068550690592160880796409808856278344061479070662988982022120971272262901\ 43887, 21350618352576141207111641167016872870072159968924190037934072466624933\ 721828950793617198312005966318224438455012306353088351467665386569169536531037\ 5600, 105766549169746657457177495160795053147675316045053760028136946076739123\ 096121747237393614122594953862859929933317474540032033044020511604290911279757\ 8000, 458962000366624281978286228136614376734271807491255555185339411606357295\ 326775587709161320577147049289132517317734572631445496019640312070175688364663\ 2400, 174876985718348094395296779940342011583608045875303144465231803155215725\ 797826520741138331953345630190089926112128587217362702723561649788589905927518\ 29488, 58632506868535284197864012527661680555969225968971476720760709101013104\ 954161595328151949665795567598109421553092637857062614656841668919340699466745\ 799432, 1733046006960190036947080477897040154187805339315811316118473068892206\ 483479346910930735595020370676519080821986640183476069166917580772512904336437\ 56455000, 4523391629574072123598739575864892042213619701872420472157378120684961\ 069465370803113061124571076369670774691598823010078890919174848898212667739790\ 58139800, 1044056812167222693213490195289089249218789792629733330159783777351773\ 132557846520169008177700085240096216159326341548192554972582519245620730582202\ 330457800, 2133659568179133056198531477079738054543140888995393841485062654567087\ 947567955693567778718574920600283429653930772833945736472530443073737301814087\ 147117968, 3864724212619131723058255840324898394917264833312827632002529241287003\ 968331132825541360683983671507811174384468555244536455978852424902735363462308\ 764842672, 6209766265138757031692206380466642904337805595264463828594452194333794\ 240089619809375451506488963397278741057520750732933799137880502887707934955565\ 305868400, 8857045003935216894450586472078719775664055104976076611573767931624652\ 304780201614284632575249728010701244747579999860173536678873613805193070301329\ 982683600, 1121960374351240067767588252609039385367649268198873226242962769683379\ 529513773746115394861772564342301297254863985653771503698208976913251693181508\ 1824648700, 1262658304856573175870480390361327360819567718997510410035097385644956\ 413335675738685242561411701781988994198892663800433580397481724942125801963269\ 4790596628, 1262658304856573175870480390361327360819567718997510410035097385644956\ 413335675738685242561411701781988994198892663800433580397481724942125801963269\ 4790596628, 1121960374351240067767588252609039385367649268198873226242962769683379\ 529513773746115394861772564342301297254863985653771503698208976913251693181508\ 1824648700, 8857045003935216894450586472078719775664055104976076611573767931624652\ 304780201614284632575249728010701244747579999860173536678873613805193070301329\ 982683600, 6209766265138757031692206380466642904337805595264463828594452194333794\ 240089619809375451506488963397278741057520750732933799137880502887707934955565\ 305868400, 3864724212619131723058255840324898394917264833312827632002529241287003\ 968331132825541360683983671507811174384468555244536455978852424902735363462308\ 764842672, 2133659568179133056198531477079738054543140888995393841485062654567087\ 947567955693567778718574920600283429653930772833945736472530443073737301814087\ 147117968, 1044056812167222693213490195289089249218789792629733330159783777351773\ 132557846520169008177700085240096216159326341548192554972582519245620730582202\ 330457800, 4523391629574072123598739575864892042213619701872420472157378120684961\ 069465370803113061124571076369670774691598823010078890919174848898212667739790\ 58139800, 1733046006960190036947080477897040154187805339315811316118473068892206\ 483479346910930735595020370676519080821986640183476069166917580772512904336437\ 56455000, 5863250686853528419786401252766168055596922596897147672076070910101310\ 495416159532815194966579556759810942155309263785706261465684166891934069946674\ 5799432, 174876985718348094395296779940342011583608045875303144465231803155215\ 725797826520741138331953345630190089926112128587217362702723561649788589905927\ 51829488, 4589620003666242819782862281366143767342718074912555551853394116063572\ 953267755877091613205771470492891325173177345726314454960196403120701756883646\ 632400, 1057665491697466574571774951607950531476753160450537600281369460767391\ 230961217472373936141225949538628599299333174745400320330440205116042909112797\ 578000, 2135061835257614120711164116701687287007215996892419003793407246662493\ 372182895079361719831200596631822443845501230635308835146766538656916953653103\ 75600, 37653186594396314358430187770167124233082550387266532535098287163511821\ 352406855069059216088079640980885627834406147907066298898202212097127226290143\ 887, 5783958852611900500094846778832172827761071302195821892407110329631433669\ 925107800048632066224360824191738573591419994783728832374141715877422664239893\ , 7713208833366058765223589370735068919325286534067549945585494851907420839776\ 01502359323391252794334455306207398111271964668528850299255764995728801125, 8896548614175103638317523790954092398174551708926790096824133998911109\ 510985320298253424796976950355956090019694879117847587438701425503087071944057\ 5, 883878610061635987757249216058013715536766253899438008883562019006307971799\ 6653138625909187604552635611061015612572287746402532786770273471667570200, 7529274154608035945139311016539314061613225250196341658798063167971160\ 59372241703956965805228964984768698298851491888082626288552627550600473893832, 547116420331901909723349056118168887881639961163335466778419196553670\ 26524232737584032137929760642880732891813207132422147203982304910314713964328, 337203255921355465002485662403928134271504860929431424869695917653081\ 8243949586995073920267657290019129425034158505367599881269650429863219484600, 1751523092780372420037392481414225382720886942337049800768668033681647\ 18897347085655765475155895729607303354564883751571083603776209114097214900, 7612801432894179060252070936899790390777621082433232295709447228594990\ 379248165429760777552405729188621973277839721881390779769352798204394300, 2746537571058434097542758583625508422589797986454737057563063064371364\ 06137582198338154436129694200720001956860133256892841199518152685889372, 8150824546136148616838063715888742369492116140083124770943734587604463\ 215938059632640592410303951676068053801339915785064711989279853247124, 1969407935454903266987622805820132250552027125764632209536159490063766\ 52354493619501386648099948903032752793954480150261445037100626798600, 3829313374254793945393686572872334037182423521347185815710848147316010\ 174976564187250083011477404277144933543825476992954269276683051800, 5912450546553281673415108580041245167893563610676610118388053901121053\ 8335101021378668641675343335571208341611366995861223686581726200, 7138761687472666571411105415389970035579498329262607471854256069932262\ 20440282861801332577837896896040137322489558074166991304308776, 6622084080973980153300161844866868346406701130858376047348832498490197\ 172507009452337815250315950214208576857197709790453711306994, 4623023305873988620338307496222990721147100351616962061036502633907059\ 5391022410144087253402634530693944178694912872915408190150, 2370862688731155069957492156947758615844535959339141508882962894276614\ 45144076007836724336676029159920342032633970952598390150, 8679502360625323483069583782389909731148220313856078961483692653965616\ 73653789388362086147831183169956746595666830985395250, 2191919217821795878036652216116897939049483674485058193042481726369136\ 187949904220388091210344135696891856845240898552456, 3663876168394461890949789100335320842345044541933916259930698386450258\ 582031719079944326775858869023957119637510836504, 3853932625773480706467888754067239916592414636326475027477521556687152\ 958678213588039313018209582087340158031478200, 2396383212206219510961374005313485605706689120200069583135463446375786\ 275983889564736797434093413137302987297000, 8143117825006705724492728039882868840142067757695981503026401567722282\ 14306753737182642757381264520835451300, 1367974644094076856258809476999389989399309382581099113029176114507149\ 61411386431312095036567719572361996, 9973183273616194294572868830073746782963597444602979734350359960442566\ 628593954317781079447309908844, 2654082645762624843326983774824319506362911611758013689381475727631009\ 07375497359328820538770500, 2036709297506271719749985964964325484971573914863649492368437972176471\ 388474497937125232600, 3234410524483621959628996874482869657908876416563072839670223548720277\ 070241422750600, 6533178267505647479118902233254345418419349219016778099576997616123116\ 31852456, 7888608889909375586561924302786347980754797673629685343360766755479224\ , 1606938044206937145948028954308115559568433722798231162561425, 515377520732011203003750506714451721535083784075, 1267650600228229401496703205275, 1}]]], "Output", FontSize->6] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(MemoryInUse[]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(5182036\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(First[Timing[l[200]]]\)], "Input"], Cell[BoxData[ \(12.370000000000001`\ Second\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(MemoryInUse[]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(11056764\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(First[Timing[l[300]]]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(24.440000000000005`\ Second\)], "Output"] }, Closed]], Cell["\<\ But such large numbers only have a meaning in a logaritmic scale:\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Last[Timing[N[Log[l[300]]]]] // TableForm\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[GridBox[{ { InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]]}, {"0.`"}, {"207.94415416798358`"}, {"329.5836866004329`"}, {"415.88830833596717`"}, {"482.8313737302301`"}, {"537.5278407684165`"}, {"583.773044716594`"}, {"623.8324625039508`"}, {"659.1673732008657`"}, {"690.7755278982081`"}, {"719.3685818393961`"}, {"745.4719949350133`"}, {"769.4848072272425`"}, {"791.7171988180255`"}, {"812.4150600219183`"}, {"831.7766154971731`"}, {"849.9639994159735`"}, {"867.1115166167594`"}, {"883.3316665704414`"}, {"898.7196195984616`"}, {"913.3565989462862`"}, {"927.3124744429841`"}, {"940.6477782942932`"}, {"953.415290830562`"}, {"965.6613018057957`"}, {"977.4266238646246`"}, {"988.7474145520895`"}, {"999.6558488742277`"}, {"1010.1806740714788`"}, {"1020.3476707184986`"}, {"1030.1800386914754`"}, {"1039.6987223865415`"}, {"1048.9226864413006`"}, {"1057.869150832072`"}, {"1066.5537923969598`"}, {"1074.9909184283306`"}, {"1083.19361688495`"}, {"1091.1738869181595`"}, {"1098.9427527319012`"}, {"1106.510363261064`"}, {"1113.8860797249604`"}, {"1121.0785527687924`"}, {"1128.0957906275294`"}, {"1134.945219519789`"}, {"1141.633737293361`"}, {"1148.167761190658`"}, {"1154.553270475202`"}, {"1160.7958445541967`"}, {"1166.9006971433555`"}, {"1172.872706945307`"}, {"1178.716445249593`"}, {"1184.436200808502`"}, {"1190.0360022971447`"}, {"1195.5196386269533`"}, {"1200.89067734812`"}, {"1206.1524813475019`"}, {"1211.3082240234876`"}, {"1216.3609030976515`"}, {"1221.313353204211`"}, {"1226.1682573819407`"}, {"1230.9281575789341`"}, {"1235.5954642681334`"}, {"1240.172465260644`"}, {"1244.661333794278`"}, {"1249.0641359663562`"}, {"1253.382837572407`"}, {"1257.6193104058682`"}, {"1261.7753380681334`"}, {"1265.8526213331916`"}, {"1269.8527831065894`"}, {"1273.7773730144386`"}, {"1277.6278716546353`"}, {"1281.4056945392845`"}, {"1285.1121957545056`"}, {"1288.7486713612755`"}, {"1292.3163625587167`"}, {"1295.8164586292226`"}, {"1299.2500996830163`"}, {"1302.6183792181155`"}, {"1305.922346510229`"}, {"1309.163008845807`"}, {"1312.3413336102847`"}, {"1315.458250242512`"}, {"1318.514652065398`"}, {"1321.5113980019405`"}, {"1324.4493141850305`"}, {"1327.3291954687145`"}, {"1330.151806847961`"}, {"1332.9178847933945`"}, {"1335.628138506931`"}, {"1338.2832511037798`"}, {"1340.8838807258269`"}, {"1343.430661591031`"}, {"1345.9242049830918`"}, {"1348.3651001853282`"}, {"1350.7539153623911`"}, {"1353.0911983931753`"}, {"1355.3774776580271`"}, {"1357.6132627831212`"}, {"1359.7990453446696`"}, {"1361.9352995354245`"}, {"1364.0224827957627`"}, {"1366.0610364114762`"}, {"1368.0513860802369`"}, {"1369.99394244857`"}, {"1371.8891016210378`"}, {"1373.7372456432217`"}, {"1375.5387429599768`"}, {"1377.2939488503328`"}, {"1379.0032058403285`"}, {"1380.6668440949688`"}, {"1382.2851817904245`"}, {"1383.8585254675131`"}, {"1385.3871703674356`"}, {"1386.8714007506746`"}, {"1388.3114901999077`"}, {"1389.707701907724`"}, {"1391.0602889498914`"}, {"1392.369494544867`"}, {"1393.6355523001994`"}, {"1394.8586864464348`"}, {"1396.0391120590907`"}, {"1397.1770352692365`"}, {"1398.2726534631724`"}, {"1399.3261554716812`"}, {"1400.3377217492823`"}, {"1401.307524543903`"}, {"1402.235728057343`"}, {"1403.1224885968977`"}, {"1403.9679547184628`"}, {"1404.772267361445`"}, {"1405.5355599757595`"}, {"1406.2579586411882`"}, {"1406.9395821793562`"}, {"1407.580542258552`"}, {"1408.1809434916172`"}, {"1408.7408835271044`"}, {"1409.2604531338866`"}, {"1409.7397362793965`"}, {"1410.1788102016474`"}, {"1410.5777454751824`"}, {"1410.9366060710836`"}, {"1411.2554494111594`"}, {"1411.5343264164155`"}, {"1411.7732815499085`"}, {"1411.9723528540592`"}, {"1412.1315719825063`"}, {"1412.2509642265545`"}, {"1412.3305485362744`"}, {"1412.3703375362907`"}, {"1412.3703375362907`"}, {"1412.3305485362744`"}, {"1412.2509642265545`"}, {"1412.1315719825063`"}, {"1411.9723528540592`"}, {"1411.7732815499085`"}, {"1411.5343264164155`"}, {"1411.2554494111594`"}, {"1410.9366060710836`"}, {"1410.5777454751824`"}, {"1410.1788102016474`"}, {"1409.7397362793965`"}, {"1409.2604531338866`"}, {"1408.7408835271044`"}, {"1408.1809434916172`"}, {"1407.580542258552`"}, {"1406.9395821793562`"}, {"1406.2579586411882`"}, {"1405.5355599757595`"}, {"1404.772267361445`"}, {"1403.9679547184628`"}, {"1403.1224885968977`"}, {"1402.235728057343`"}, {"1401.307524543903`"}, {"1400.3377217492823`"}, {"1399.3261554716812`"}, {"1398.2726534631724`"}, {"1397.1770352692365`"}, {"1396.0391120590907`"}, {"1394.8586864464348`"}, {"1393.6355523001994`"}, {"1392.369494544867`"}, {"1391.0602889498914`"}, {"1389.707701907724`"}, {"1388.3114901999077`"}, {"1386.8714007506746`"}, {"1385.3871703674356`"}, {"1383.8585254675131`"}, {"1382.2851817904245`"}, {"1380.6668440949688`"}, {"1379.0032058403285`"}, {"1377.2939488503328`"}, {"1375.5387429599768`"}, {"1373.7372456432217`"}, {"1371.8891016210378`"}, {"1369.99394244857`"}, {"1368.0513860802369`"}, {"1366.0610364114762`"}, {"1364.0224827957627`"}, {"1361.9352995354245`"}, {"1359.7990453446696`"}, {"1357.6132627831212`"}, {"1355.3774776580271`"}, {"1353.0911983931753`"}, {"1350.7539153623911`"}, {"1348.3651001853282`"}, {"1345.9242049830918`"}, {"1343.430661591031`"}, {"1340.8838807258269`"}, {"1338.2832511037798`"}, {"1335.628138506931`"}, {"1332.9178847933945`"}, {"1330.151806847961`"}, {"1327.3291954687145`"}, {"1324.4493141850305`"}, {"1321.5113980019405`"}, {"1318.514652065398`"}, {"1315.458250242512`"}, {"1312.3413336102847`"}, {"1309.163008845807`"}, {"1305.922346510229`"}, {"1302.6183792181155`"}, {"1299.2500996830163`"}, {"1295.8164586292226`"}, {"1292.3163625587167`"}, {"1288.7486713612755`"}, {"1285.1121957545056`"}, {"1281.4056945392845`"}, {"1277.6278716546353`"}, {"1273.7773730144386`"}, {"1269.8527831065894`"}, {"1265.8526213331916`"}, {"1261.7753380681334`"}, {"1257.6193104058682`"}, {"1253.382837572407`"}, {"1249.0641359663562`"}, {"1244.661333794278`"}, {"1240.172465260644`"}, {"1235.5954642681334`"}, {"1230.9281575789341`"}, {"1226.1682573819407`"}, {"1221.313353204211`"}, {"1216.3609030976515`"}, {"1211.3082240234876`"}, {"1206.1524813475019`"}, {"1200.89067734812`"}, {"1195.5196386269533`"}, {"1190.0360022971447`"}, {"1184.436200808502`"}, {"1178.716445249593`"}, {"1172.872706945307`"}, {"1166.9006971433555`"}, {"1160.7958445541967`"}, {"1154.553270475202`"}, {"1148.167761190658`"}, {"1141.633737293361`"}, {"1134.945219519789`"}, {"1128.0957906275294`"}, {"1121.0785527687924`"}, {"1113.8860797249604`"}, {"1106.510363261064`"}, {"1098.9427527319012`"}, {"1091.1738869181595`"}, {"1083.19361688495`"}, {"1074.9909184283306`"}, {"1066.5537923969598`"}, {"1057.869150832072`"}, {"1048.9226864413006`"}, {"1039.6987223865415`"}, {"1030.1800386914754`"}, {"1020.3476707184986`"}, {"1010.1806740714788`"}, {"999.6558488742277`"}, {"988.7474145520895`"}, {"977.4266238646246`"}, {"965.6613018057957`"}, {"953.415290830562`"}, {"940.6477782942932`"}, {"927.3124744429841`"}, {"913.3565989462862`"}, {"898.7196195984616`"}, {"883.3316665704414`"}, {"867.1115166167594`"}, {"849.9639994159735`"}, {"831.7766154971731`"}, {"812.4150600219183`"}, {"791.7171988180255`"}, {"769.4848072272425`"}, {"745.4719949350133`"}, {"719.3685818393961`"}, {"690.7755278982081`"}, {"659.1673732008657`"}, {"623.8324625039508`"}, {"583.773044716594`"}, {"537.5278407684165`"}, {"482.8313737302301`"}, {"415.88830833596717`"}, {"329.5836866004329`"}, {"207.94415416798358`"}, {"0.`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { DirectedInfinity[ -1], 0.0, 207.94415416798358, 329.58368660043288, 415.88830833596717, 482.8313737302301, 537.52784076841647, 583.77304471659397, 623.83246250395075, 659.16737320086565, 690.77552789820811, 719.36858183939614, 745.4719949350133, 769.48480722724253, 791.71719881802551, 812.41506002191829, 831.77661549717311, 849.96399941597349, 867.11151661675945, 883.33166657044137, 898.71961959846158, 913.35659894628623, 927.31247444298413, 940.64777829429318, 953.41529083056196, 965.66130180579569, 977.42662386462462, 988.74741455208948, 999.65584887422767, 1010.1806740714788, 1020.3476707184986, 1030.1800386914754, 1039.6987223865415, 1048.9226864413006, 1057.8691508320719, 1066.5537923969598, 1074.9909184283306, 1083.19361688495, 1091.1738869181595, 1098.9427527319012, 1106.510363261064, 1113.8860797249604, 1121.0785527687924, 1128.0957906275294, 1134.9452195197889, 1141.633737293361, 1148.167761190658, 1154.553270475202, 1160.7958445541967, 1166.9006971433555, 1172.8727069453071, 1178.7164452495931, 1184.436200808502, 1190.0360022971447, 1195.5196386269533, 1200.8906773481201, 1206.1524813475019, 1211.3082240234876, 1216.3609030976515, 1221.3133532042109, 1226.1682573819407, 1230.9281575789341, 1235.5954642681334, 1240.172465260644, 1244.6613337942781, 1249.0641359663562, 1253.382837572407, 1257.6193104058682, 1261.7753380681334, 1265.8526213331916, 1269.8527831065894, 1273.7773730144386, 1277.6278716546353, 1281.4056945392845, 1285.1121957545056, 1288.7486713612755, 1292.3163625587167, 1295.8164586292226, 1299.2500996830163, 1302.6183792181155, 1305.9223465102291, 1309.1630088458071, 1312.3413336102847, 1315.4582502425119, 1318.514652065398, 1321.5113980019405, 1324.4493141850305, 1327.3291954687145, 1330.1518068479611, 1332.9178847933945, 1335.628138506931, 1338.2832511037798, 1340.8838807258269, 1343.4306615910309, 1345.9242049830918, 1348.3651001853282, 1350.7539153623911, 1353.0911983931753, 1355.3774776580271, 1357.6132627831212, 1359.7990453446696, 1361.9352995354245, 1364.0224827957627, 1366.0610364114762, 1368.0513860802369, 1369.9939424485699, 1371.8891016210378, 1373.7372456432217, 1375.5387429599768, 1377.2939488503328, 1379.0032058403285, 1380.6668440949688, 1382.2851817904245, 1383.8585254675131, 1385.3871703674356, 1386.8714007506746, 1388.3114901999077, 1389.707701907724, 1391.0602889498914, 1392.3694945448669, 1393.6355523001994, 1394.8586864464348, 1396.0391120590907, 1397.1770352692365, 1398.2726534631724, 1399.3261554716812, 1400.3377217492823, 1401.3075245439029, 1402.2357280573431, 1403.1224885968977, 1403.9679547184628, 1404.7722673614451, 1405.5355599757595, 1406.2579586411882, 1406.9395821793562, 1407.580542258552, 1408.1809434916172, 1408.7408835271044, 1409.2604531338866, 1409.7397362793965, 1410.1788102016474, 1410.5777454751824, 1410.9366060710836, 1411.2554494111594, 1411.5343264164155, 1411.7732815499085, 1411.9723528540592, 1412.1315719825063, 1412.2509642265545, 1412.3305485362744, 1412.3703375362907, 1412.3703375362907, 1412.3305485362744, 1412.2509642265545, 1412.1315719825063, 1411.9723528540592, 1411.7732815499085, 1411.5343264164155, 1411.2554494111594, 1410.9366060710836, 1410.5777454751824, 1410.1788102016474, 1409.7397362793965, 1409.2604531338866, 1408.7408835271044, 1408.1809434916172, 1407.580542258552, 1406.9395821793562, 1406.2579586411882, 1405.5355599757595, 1404.7722673614451, 1403.9679547184628, 1403.1224885968977, 1402.2357280573431, 1401.3075245439029, 1400.3377217492823, 1399.3261554716812, 1398.2726534631724, 1397.1770352692365, 1396.0391120590907, 1394.8586864464348, 1393.6355523001994, 1392.3694945448669, 1391.0602889498914, 1389.707701907724, 1388.3114901999077, 1386.8714007506746, 1385.3871703674356, 1383.8585254675131, 1382.2851817904245, 1380.6668440949688, 1379.0032058403285, 1377.2939488503328, 1375.5387429599768, 1373.7372456432217, 1371.8891016210378, 1369.9939424485699, 1368.0513860802369, 1366.0610364114762, 1364.0224827957627, 1361.9352995354245, 1359.7990453446696, 1357.6132627831212, 1355.3774776580271, 1353.0911983931753, 1350.7539153623911, 1348.3651001853282, 1345.9242049830918, 1343.4306615910309, 1340.8838807258269, 1338.2832511037798, 1335.628138506931, 1332.9178847933945, 1330.1518068479611, 1327.3291954687145, 1324.4493141850305, 1321.5113980019405, 1318.514652065398, 1315.4582502425119, 1312.3413336102847, 1309.1630088458071, 1305.9223465102291, 1302.6183792181155, 1299.2500996830163, 1295.8164586292226, 1292.3163625587167, 1288.7486713612755, 1285.1121957545056, 1281.4056945392845, 1277.6278716546353, 1273.7773730144386, 1269.8527831065894, 1265.8526213331916, 1261.7753380681334, 1257.6193104058682, 1253.382837572407, 1249.0641359663562, 1244.6613337942781, 1240.172465260644, 1235.5954642681334, 1230.9281575789341, 1226.1682573819407, 1221.3133532042109, 1216.3609030976515, 1211.3082240234876, 1206.1524813475019, 1200.8906773481201, 1195.5196386269533, 1190.0360022971447, 1184.436200808502, 1178.7164452495931, 1172.8727069453071, 1166.9006971433555, 1160.7958445541967, 1154.553270475202, 1148.167761190658, 1141.633737293361, 1134.9452195197889, 1128.0957906275294, 1121.0785527687924, 1113.8860797249604, 1106.510363261064, 1098.9427527319012, 1091.1738869181595, 1083.19361688495, 1074.9909184283306, 1066.5537923969598, 1057.8691508320719, 1048.9226864413006, 1039.6987223865415, 1030.1800386914754, 1020.3476707184986, 1010.1806740714788, 999.65584887422767, 988.74741455208948, 977.42662386462462, 965.66130180579569, 953.41529083056196, 940.64777829429318, 927.31247444298413, 913.35659894628623, 898.71961959846158, 883.33166657044137, 867.11151661675945, 849.96399941597349, 831.77661549717311, 812.41506002191829, 791.71719881802551, 769.48480722724253, 745.4719949350133, 719.36858183939614, 690.77552789820811, 659.16737320086565, 623.83246250395075, 583.77304471659397, 537.52784076841647, 482.8313737302301, 415.88830833596717, 329.58368660043288, 207.94415416798358, 0.0}]]], "Output"] }, Closed]], Cell["\<\ Writing these coefficients out would put some 150000 digits on the \ screen.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Assignments", "Subsection"], Cell["Look for EulerE in Help.", "Subsubsection"], Cell["Benchmark your computer by comparing the above timings.", \ "Subsubsection"] }, Closed]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.1 for Macintosh", ScreenRectangle->{{0, 1152}, {0, 746}}, WindowSize->{857, 670}, WindowMargins->{{30, Automatic}, {Automatic, 18}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1727, 52, 101, 1, 65, "Subtitle"], Cell[1831, 55, 83, 4, 89, "Subsubtitle"], Cell[CellGroupData[{ Cell[1939, 63, 37, 0, 56, "Section"], Cell[1979, 65, 74, 0, 30, "Text"], Cell[2056, 67, 199, 4, 61, "Input", Evaluatable->False], Cell[2258, 73, 50, 0, 30, "Text"], Cell[2311, 75, 148, 3, 45, "Input", Evaluatable->False], Cell[2462, 80, 231, 5, 46, "Text"], Cell[2696, 87, 145, 3, 51, "Input", Evaluatable->False], Cell[2844, 92, 260, 6, 48, "Text"], Cell[3107, 100, 40, 0, 30, "Text"], Cell[CellGroupData[{ Cell[3172, 104, 77, 0, 27, "Input"], Cell[3252, 106, 6760, 217, 401, "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[10061, 329, 89, 1, 36, "Section"], Cell[10153, 332, 136, 4, 30, "Text"], Cell[10292, 338, 72, 2, 27, "Input"], Cell[10367, 342, 86, 2, 27, "Input"], Cell[10456, 346, 108, 2, 27, "Input"], Cell[10567, 350, 172, 5, 27, "Input"], Cell[CellGroupData[{ Cell[10764, 359, 65, 2, 27, "Input"], Cell[10832, 363, 51, 1, 27, "Output"] }, Closed]], Cell[10898, 367, 93, 1, 27, "Text"], Cell[CellGroupData[{ Cell[11016, 372, 74, 2, 27, "Input"], Cell[11093, 376, 143, 2, 27, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[11273, 383, 81, 2, 24, "Input"], Cell[11357, 387, 509, 19, 197, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[11903, 411, 74, 2, 24, "Input"], Cell[11980, 415, 41, 1, 27, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[12058, 421, 74, 2, 24, "Input"], Cell[12135, 425, 158, 2, 27, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[12330, 432, 74, 2, 24, "Input"], Cell[12407, 436, 41, 1, 27, "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[12497, 443, 86, 1, 36, "Section"], Cell[CellGroupData[{ Cell[12608, 448, 45, 0, 46, "Subsection"], Cell[12656, 450, 164, 4, 30, "Text"], Cell[12823, 456, 72, 2, 27, "Input"], Cell[12898, 460, 86, 2, 27, "Input"], Cell[12987, 464, 118, 2, 27, "Input"], Cell[13108, 468, 154, 4, 27, "Input"], Cell[CellGroupData[{ Cell[13287, 476, 65, 2, 27, "Input"], Cell[13355, 480, 51, 1, 27, "Output"] }, Closed]], Cell[13421, 484, 78, 1, 27, "Text"], Cell[CellGroupData[{ Cell[13524, 489, 74, 2, 27, "Input"], Cell[13601, 493, 145, 2, 27, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[13783, 500, 74, 2, 24, "Input"], Cell[13860, 504, 41, 1, 27, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[13938, 510, 81, 2, 24, "Input"], Cell[14022, 514, 61, 1, 27, "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[14132, 521, 90, 1, 30, "Subsection"], Cell[CellGroupData[{ Cell[14247, 526, 93, 2, 27, "Input"], Cell[14343, 530, 5224, 91, 757, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[19604, 626, 74, 2, 27, "Input"], Cell[19681, 630, 41, 1, 27, "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[19771, 637, 74, 1, 30, "Subsection"], Cell[19848, 640, 121, 4, 30, "Text"], Cell[CellGroupData[{ Cell[19994, 648, 94, 2, 27, "Input"], Cell[20091, 652, 29786, 548, 831, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[49914, 1205, 74, 2, 27, "Input"], Cell[49991, 1209, 41, 1, 27, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[50069, 1215, 54, 1, 27, "Input"], Cell[50126, 1218, 61, 1, 27, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[50224, 1224, 74, 2, 27, "Input"], Cell[50301, 1228, 42, 1, 27, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[50380, 1234, 82, 2, 27, "Input"], Cell[50465, 1238, 61, 1, 27, "Output"] }, Closed]], Cell[50541, 1242, 91, 3, 30, "Text"], Cell[CellGroupData[{ Cell[50657, 1249, 102, 2, 27, "Input"], Cell[50762, 1253, 17520, 410, 4837, "Output"] }, Closed]], Cell[68297, 1666, 100, 3, 30, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[68434, 1674, 33, 0, 30, "Subsection"], Cell[68470, 1676, 49, 0, 42, "Subsubsection"], Cell[68522, 1678, 82, 1, 42, "Subsubsection"] }, Closed]] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)