2018年9月24日月曜日

180924

Ruby


Expansion of theta_4(q)^b in powers of q = exp(Pi i t)

-28 <= b <= 28 のとき、出力してみた。

def s(k, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == 0}
  s
end

def A(ary, n)
  a_ary = [1]
  a = [0] + (1..n).map{|i| ary.inject(0){|s, j| s + j[1] * s(j[0], i)}}
  (1..n).each{|i| a_ary << (1..i).inject(0){|s, j| s - a[j] * a_ary[-j]} / i}
  a_ary
end

def B(k, n)
  A([[1, 2 * k], [2, -k]], n)
end

n = 30
-28.upto(28){|i| p [i, B(i, n)]}

出力結果
[-28, [1, 56, 1624, 32480, 503384, 6440784, 70786912, 686577088, 5992951384, 47776150744, 351872947216, 2416267224224, 15585879310176, 95023751851280, 550435283089088, 3042813237891392, 16113562252291672, 82014156269030384, 402366072890195064, 1907630974993082976, 8759675533443400720, 39037020391980812032, 169139235544857978400, 713670307390583841088, 2936810077908914847584, 11802121500158402988232, 46374603353250996050640, 178369757578402864153792, 672246246931518841059008, 2484915984178103534196880, 9016744845844611900634176]]
[-27, [1, 54, 1512, 29232, 438426, 5434128, 57908592, 545068224, 4620803976, 35803129542, 256462219440, 1713894480720, 10765330699536, 63947544525648, 361091512399392, 1946775064587840, 10059106161975066, 49977067282246944, 239438535366014232, 1108983751942646160, 4976635875428376432, 21681666465886332576, 91869736859314615728, 379205997381661026240, 1526982728580144086832, 6006543158095582635270, 23108474796415113864528, 87047086759368793789760, 321376536699752714836032, 1164010214154363273949680, 4139584011254993416014240]]
[-26, [1, 52, 1404, 26208, 379964, 4557384, 47043360, 429309504, 3531544380, 26572389844, 184971777432, 1202060175264, 7346783287584, 42488488508808, 233709336918720, 1228019712933312, 6187085604062012, 29986740805740264, 140206512220063068, 633999718415412960, 2778783156078949656, 11828327318943542784, 48985153103300744928, 197683768984982424768, 778520112026791219488, 2995920915726349435708, 11279028000320937770328, 41588070287658945043136, 150334110692202191478464, 533260273927193240969736, 1857737169928348337109312]]
[-25, [1, 50, 1300, 23400, 327550, 3797560, 37931400, 335275200, 2673685300, 19518415850, 131921084360, 832975236600, 4949725973400, 27848065953400, 149102294341200, 763005065359680, 3745738886151550, 17697539511545200, 80700396137643100, 356042363457651000, 1523155486336657160, 6330719794839663600, 25608826512970854600, 100980994797396542400, 388707684336621491400, 1462524972192590737962, 5385096501963172463000, 19425101414366714573600, 68713742356194619528000, 238578336892906375242600, 813751021887384508347600]]
[-24, [1, 48, 1200, 20800, 280752, 3142560, 30338880, 259459200, 2003790000, 14178640368, 92960115360, 569803615680, 3289122824000, 17987650183200, 93669997008000, 466466351287680, 2229627536828592, 10261752523778400, 45603482666406000, 196167317470962240, 818560571243840160, 3319794001745088000, 13108732019510769600, 50475128294452003200, 189790479709718372160, 697763551670291610960, 2511228815788751774496, 8856709000908837916800, 30640242821306969961600, 104072855869625775599904, 347351879964113004548480]]
[-23, [1, 46, 1104, 18400, 239154, 2581152, 24056160, 198823040, 1485433104, 10177345486, 64663512288, 384402300960, 2153523131040, 11437761254432, 57880610587200, 280265903825280, 1303272560982834, 5838468742907712, 25267289255586864, 105893098920536480, 430684051768903008, 1703189786065601600, 6560315006113716960, 24649811550897290880, 90476411334607050720, 324817721027628666190, 1141898653168062765600, 3935110420094512791168, 13306013450852034554240, 44186260245185231023200, 144221986455775454452544]]
[-22, [1, 44, 1012, 16192, 202356, 2102936, 18896064, 150747520, 1088265332, 7211641580, 44356933544, 255472920256, 1387689358528, 7151069205016, 35134409940608, 165273439140480, 747047401948276, 3254796172584792, 13706092636257076, 55918922890025536, 221503630245672104, 853494312972511232, 3204447137019290944, 11740851975687401600, 42037595807513490624, 147268671405778493252, 505372164316874076200, 1700562203050475267712, 5616533651441428225664, 18223007440308403471640, 58129857007930202746240]]
[-21, [1, 42, 924, 14168, 169974, 1698312, 14692216, 112987776, 787175004, 5039316786, 29971442424, 167060546184, 878920016296, 4390113366408, 20920981191792, 95515527307648, 419275600889334, 1775001330567696, 7266754687687572, 28837195781957640, 111159368324081784, 416993869451825424, 1524856394246164728, 5443725278910484608, 18998568571928281336, 64898974336315921074, 217237206275724369000, 713272166693775242848, 2299372601885103305088, 7284034487278750667928, 22692879537590014608240]]
[-20, [1, 40, 840, 12320, 141640, 1358448, 11297440, 83631680, 561539400, 3468363400, 19922193200, 107343635040, 546373245600, 2642351627440, 12200693947200, 54007656632000, 230002160331080, 945228781171920, 3758583910304040, 14494657518341280, 54323193841192752, 198223377628161280, 705393384520540640, 2451645040245662400, 8333245721275210400, 27734986390239864152, 90485420448866605680, 289670671524893189440, 910767733007685471040, 2814864922224754129200, 8558451029687017346496]]
[-19, [1, 38, 760, 10640, 117002, 1075248, 8582224, 61061440, 394559320, 2348001494, 13008061200, 67666510320, 332809029680, 1556541579760, 6955832361824, 29820933412800, 123079426294922, 490508040685920, 1892520799129480, 7085465166635440, 25793668438185552, 91466354764345312, 316458512404166160, 1069823185020632640, 3538494014276085904, 11464481705561830550, 36424344019710001136, 113595603754517963200, 348063893178159296960, 1048690787886883501520, 3109295603781371154400]]
[-18, [1, 36, 684, 9120, 95724, 841320, 6433248, 43918272, 272670444, 1561033348, 8329222584, 41772509280, 198265106400, 895619289384, 3868763174208, 16044584545344, 64103055405804, 247461482137032, 925411686479820, 3360039419100960, 11868761624717880, 40859368353437184, 137307731407739424, 451063179002970432, 1450375340033484768, 4570180232381762508, 14127316643329422840, 42882777178873986368, 127935503432960309568, 375440380717895123112, 1084579797249158036160]]
[-17, [1, 34, 612, 7752, 77486, 649944, 4751976, 31070016, 185025348, 1017375098, 5220022312, 25201899288, 115265410488, 502210951832, 2094181357968, 8390590348992, 32410328691374, 121046064563376, 438223126488844, 1541288556805080, 5276805352500456, 17616371539745328, 57438026848112424, 183160057587181632, 571954416653229864, 1751020086032922426, 5261076735075547704, 15528419510393139104, 45063980072971299008, 128685998584726980552, 361871958100844427024]]
[-16, [1, 32, 544, 6528, 61984, 495040, 3453312, 21581568, 123040288, 648624288, 3194776000, 14823993472, 65231647104, 273714726080, 1100198199040, 4252621927680, 15859616674336, 57229459033664, 200317325329056, 681613044496256, 2258998492629440, 7304622434350080, 23080725609199744, 71362392703991040, 216171131373220224, 642279177183320800, 1873670535016845504, 5371709759567226624, 15147967538908675840, 42049526004566975168, 114986885846823486720]]
[-15, [1, 30, 480, 5440, 48930, 371136, 2464320, 14688000, 80001120, 403533790, 1904433984, 8477603520, 35829727680, 144548556480, 559157308800, 2081866609920, 7484792950050, 26057409056640, 88057506412320, 289477248154560, 927461989122624, 2900953904854400, 8871567968612160, 26561967221349120, 77955837952722240, 224514441369480414, 635159179340088000, 1766692869938845440, 4835527186894951680, 13033631831560334400, 34620567863249935232]]
[-14, [1, 28, 420, 4480, 38052, 273336, 1723008, 9770240, 50722980, 244273820, 1102294984, 4698110592, 19034512000, 73696070840, 273868321536, 980502270720, 3392689809572, 11376760267320, 37060195850020, 117522159692672, 363460123604424, 1098073086009344, 3245486079543680, 9396667495192320, 26682663915051648, 74389704494912212, 203821321529106120, 549322347643749120, 1457474240085840640, 3809751705628161720, 9817821769313608960]]
[-13, [1, 26, 364, 3640, 29094, 197288, 1177176, 6333184, 31258604, 143374530, 617193304, 2513060264, 9739727816, 36115518376, 128680223152, 442158402816, 1469734751654, 4738671343952, 14853923411652, 45360567953064, 135191787693144, 393868290857808, 1123312910529368, 3140191938387712, 8614288268780376, 23213737945756098, 61510133382424328, 160398928580441824, 411959866378075392, 1042855861591161592, 2603765402679575472]]
[-12, [1, 24, 312, 2912, 21816, 139152, 783328, 3986112, 18650424, 81251896, 332798544, 1291339296, 4776117216, 16922753616, 57683178432, 189821722688, 604884735288, 1871370360240, 5633654421720, 16535803556064, 47405095227984, 132942579098368, 365211946954656, 984029905455168, 2603403713102816, 6769958807156712, 17319807203669520, 43629342871665088, 108299884098294720, 265093293382115280, 640286467040437056]]
[-11, [1, 22, 264, 2288, 15994, 95568, 505648, 2425280, 10721832, 44229350, 171861360, 633713808, 2230733648, 7532979344, 24502989984, 77036477760, 234785552122, 695409096096, 2006117554936, 5647472566736, 15540674951472, 41865577406752, 110560253660784, 286562260273344, 729774486503920, 1827826932181030, 4506571408141968, 10946523549070912, 26214943726286656, 61938666454846896, 144473724891427360]]
[-10, [1, 20, 220, 1760, 11420, 63624, 315040, 1418560, 5903260, 22976820, 84413912, 294841120, 984745120, 3159938760, 9780562880, 29296914112, 85169213340, 240882506920, 664216884540, 1788966694240, 4714033526616, 12170584419840, 30826269009760, 76687696955840, 187577264348320, 451539631341148, 1070647576631960, 2502494285650880, 5770155117481920, 13133366582659400, 29525852648992832]]
[-9, [1, 18, 180, 1320, 7902, 40824, 188232, 792000, 3088980, 11297546, 39090312, 128849976, 406865880, 1236379320, 3629385936, 10324840512, 28542038238, 76852151280, 201967043260, 518957929080, 1305848905416, 3222313017456, 7807214691720, 18593534036160, 43571355468168, 100555979011146, 228739255783128, 513246682211360, 1136745110975040, 2486713863179304, 5376093029096784]]
[-8, [1, 16, 144, 960, 5264, 25056, 106944, 418176, 1520784, 5201232, 16871648, 52252992, 155341248, 445226848, 1234726272, 3323392128, 8704504976, 22234655520, 55498917840, 135595345600, 324759439584, 763505859072, 1764050361152, 4009763323008, 8975341703616, 19800832628336, 43088101022304, 92551562330496, 196357355359616, 411726374183520, 853706446959744]]
[-7, [1, 14, 112, 672, 3346, 14560, 57120, 206208, 694960, 2209774, 6683040, 19345760, 53874912, 144936288, 377965760, 958231680, 2367566866, 5713057728, 13488657168, 31210552800, 70873262880, 158145658560, 347146238880, 750384675712, 1598690194080, 3359762331758, 6970100920160, 14283993601920, 28934034095232, 57964700284960, 114905358321600]]
[-6, [1, 12, 84, 448, 2004, 7896, 28224, 93312, 289236, 848972, 2377704, 6391872, 16571968, 41599320, 101430144, 240877440, 558440916, 1266406680, 2814053908, 6136337088, 13148606184, 27717527552, 57542477760, 117757784448, 237752294976, 473942546532, 933455985000, 1817622489984, 3501113922432, 6674700098136, 12600624177280]]
[-5, [1, 10, 60, 280, 1110, 3912, 12600, 37760, 106620, 286290, 736184, 1822920, 4365800, 10149320, 22971120, 50744448, 109643350, 232145040, 482403060, 985229640, 1980034104, 3920000400, 7652388280, 14742829440, 28052762040, 52757827794, 98127789480, 180610764640, 329134001280, 594143328600, 1062903740400]]
[-4, [1, 8, 40, 160, 552, 1712, 4896, 13120, 33320, 80872, 188784, 425952, 932640, 1988080, 4137024, 8422848, 16810536, 32943760, 63482760, 120440608, 225217904, 415498496, 756920160, 1362645440, 2425895712, 4273590392, 7454092720, 12879684160, 22056267840, 37451209840, 63078595776]]
[-3, [1, 6, 24, 80, 234, 624, 1552, 3648, 8184, 17654, 36816, 74544, 147056, 283440, 535008, 990912, 1803882, 3232224, 5707624, 9943536, 17106960, 29088352, 48922320, 81438528, 134261584, 219336630, 355242288, 570675904, 909674688, 1439394192, 2261635168]]
[-2, [1, 4, 12, 32, 76, 168, 352, 704, 1356, 2532, 4600, 8160, 14176, 24168, 40512, 66880, 108876, 174984, 277932, 436640, 679032, 1046016, 1597088, 2418240, 3632992, 5417708, 8022840, 11802176, 17252928, 25070568, 36223424]]
[-1, [1, 2, 4, 8, 14, 24, 40, 64, 100, 154, 232, 344, 504, 728, 1040, 1472, 2062, 2864, 3948, 5400, 7336, 9904, 13288, 17728, 23528, 31066, 40824, 53408, 69568, 90248, 116624]]
[0, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]
[1, [1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0]]
[2, [1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, 0, 0, -12, 8, 0, 0, -8, 0]]
[3, [1, -6, 12, -8, 6, -24, 24, 0, 12, -30, 24, -24, 8, -24, 48, 0, 6, -48, 36, -24, 24, -48, 24, 0, 24, -30, 72, -32, 0, -72, 48]]
[4, [1, -8, 24, -32, 24, -48, 96, -64, 24, -104, 144, -96, 96, -112, 192, -192, 24, -144, 312, -160, 144, -256, 288, -192, 96, -248, 336, -320, 192, -240, 576]]
[5, [1, -10, 40, -80, 90, -112, 240, -320, 200, -250, 560, -560, 400, -560, 800, -960, 730, -480, 1240, -1520, 752, -1120, 1840, -1600, 1200, -1210, 2000, -2240, 1600, -1680, 2720]]
[6, [1, -12, 60, -160, 252, -312, 544, -960, 1020, -876, 1560, -2400, 2080, -2040, 3264, -4160, 4092, -3480, 4380, -7200, 6552, -4608, 8160, -10560, 8224, -7812, 10200, -13120, 12480, -10104, 14144]]
[7, [1, -14, 84, -280, 574, -840, 1288, -2368, 3444, -3542, 4424, -7560, 9240, -8456, 11088, -16576, 18494, -17808, 19740, -27720, 34440, -29456, 31304, -49728, 52808, -43414, 52248, -68320, 74048, -68376, 71120]]
[8, [1, -16, 112, -448, 1136, -2016, 3136, -5504, 9328, -12112, 14112, -21312, 31808, -35168, 38528, -56448, 74864, -78624, 84784, -109760, 143136, -154112, 149184, -194688, 261184, -252016, 246176, -327040, 390784, -390240, 395136]]
[9, [1, -18, 144, -672, 2034, -4320, 7392, -12672, 22608, -34802, 44640, -60768, 93984, -125280, 141120, -182400, 262386, -317376, 343536, -421344, 557280, -665280, 703584, -800640, 1068384, -1256562, 1234080, -1421184, 1851264, -2034720, 2057280]]
[10, [1, -20, 180, -960, 3380, -8424, 16320, -28800, 52020, -88660, 129064, -175680, 262080, -386920, 489600, -600960, 840500, -1137960, 1330420, -1563840, 2050344, -2611200, 2986560, -3358080, 4194240, -5318268, 5878440, -6299520, 7862400, -9619560, 10216320]]
[11, [1, -22, 220, -1320, 5302, -15224, 33528, -63360, 116380, -209550, 339064, -491768, 719400, -1095160, 1538416, -1964160, 2624182, -3696880, 4763220, -5686648, 7217144, -9528816, 11676280, -13495680, 16317048, -20787470, 25022184, -27785120, 32503680, -40862184, 47430768]]
[12, [1, -24, 264, -1760, 7944, -25872, 64416, -133056, 253704, -472760, 825264, -1297056, 1938336, -2963664, 4437312, -6091584, 8118024, -11368368, 15653352, -19822176, 24832944, -32826112, 42517728, -51425088, 61903776, -78146664, 98021616, -115331264, 133522752, -164079696, 201364416]]
[13, [1, -26, 312, -2288, 11466, -41808, 116688, -265408, 535704, -1031914, 1899664, -3214224, 5043376, -7801744, 12066912, -17689152, 24443978, -34039200, 48210760, -64966096, 83323344, -109157152, 145532816, -185245632, 227110416, -284788010, 363737712, -448126016, 531517376, -643027632, 800145632]]
[14, [1, -28, 364, -2912, 16044, -64792, 200928, -503360, 1089452, -2186940, 4196920, -7544992, 12547808, -19975256, 31553344, -48484800, 70439852, -99602104, 142487436, -200569824, 268594872, -354052608, 476105504, -630908096, 800698080, -1008274932, 1296257144, -1648943296, 2022013760, -2457911512, 3056208064]]
[15, [1, -30, 420, -3640, 21870, -96936, 331240, -911040, 2128260, -4495430, 8972712, -16946280, 29822520, -49476840, 80027280, -127083328, 193511790, -282611280, 409172940, -590913960, 825736296, -1115671760, 1509537960, -2048372160, 2698852520, -3463029894, 4472616120, -5803339360, 7347537600, -9095661240, 11335886864]]
[16, [1, -32, 480, -4480, 29152, -140736, 525952, -1580800, 3994080, -8945824, 18626112, -36714624, 67978880, -118156480, 197120256, -321692928, 509145568, -772845120, 1143441760, -1681379200, 2428524096, -3392205824, 4658843520, -6411152640, 8705492608, -11488092896, 15000919872, -19697555200, 25593401600, -32467266240, 40864033536]]
[17, [1, -34, 544, -5440, 38114, -199104, 808384, -2641664, 7213984, -17215458, 37569728, -77129408, 149405248, -272064192, 470966912, -790414592, 1291650786, -2035437440, 3101265120, -4648909760, 6883061184, -9931183488, 13967503552, -19489000192, 27036220096, -36719672546, 48877735232, -64882013952, 85898222336, -111723674688, 142979339392]]
[18, [1, -36, 612, -6528, 48996, -275400, 1207680, -4269312, 12573540, -32041636, 73617480, -157553280, 318102912, -605381832, 1090632960, -1888224000, 3176573796, -5177295432, 8148505828, -12507419520, 18918517320, -28067389440, 40578145920, -57669043968, 81375600000, -113272212204, 154398101832, -207972274944, 279331331328, -371269791432, 484986858240]]
[19, [1, -38, 684, -7752, 62054, -373464, 1759704, -6697728, 21210156, -57739518, 140116184, -313328088, 658369608, -1305768920, 2449182384, -4387011840, 7597766246, -12766353840, 20748447108, -32728127192, 50601708504, -76895771568, 114223115480, -166029979392, 238352317656, -338575421950, 472600643976, -648718685984, 883678139136, -1195524774792, 1594639240368]]
[20, [1, -40, 760, -9120, 77560, -497648, 2508000, -10232640, 34729720, -100906760, 259114704, -606957280, 1327461600, -2738111280, 5341699520, -9915552192, 17701924600, -30615844560, 51294999960, -83279292960, 131880275664, -204949382400, 312126610080, -464844224960, 680432137440, -984382611928, 1404010658960, -1968103484480, 2724593155520, -3743846975920, 5089580732736]]
[21, [1, -42, 840, -10640, 95802, -652848, 3504816, -15266880, 55350120, -171361498, 465944304, -1145653488, 2610853840, -5595003120, 11337146784, -21817881792, 40208417466, -71565375840, 123411903160, -206218644912, 335056780464, -532597541344, 830096613360, -1266359895360, 1893781702512, -2790751314330, 4058779021584, -5809617168320, 8190877926720, -11431035439824, 15805879161632]]
[22, [1, -44, 924, -12320, 117084, -844536, 4812192, -22297792, 86075484, -283367436, 815677016, -2108077536, 5012882336, -11159625400, 23460347328, -46790491968, 89096422492, -163282917336, 289559673084, -497610124704, 830202403800, -1351064867328, 2153312518240, -3361695239232, 5140361517984, -7724269359972, 11444537517336, -16709010117184, 24015334687168, -34075597680696, 47873378766400]]
[23, [1, -46, 1012, -14168, 141726, -1078792, 6503112, -31946816, 130905236, -457207478, 1392036360, -3784029000, 9401213016, -21752359112, 47412641232, -97936966080, 192747766686, -363967164176, 663612939132, -1171941833480, 2007831887432, -3348062963344, 5456890474248, -8710505815360, 13617909938952, -20885881710454, 31527898462552, -46904811133408, 68719323996224, -99237530067544, 141662701280848]]
[24, [1, -48, 1104, -16192, 170064, -1362336, 8662720, -44981376, 195082320, -721175536, 2319457632, -6631997376, 17231109824, -41469483552, 93703589760, -200343312768, 407488018512, -793229226336, 1487286966928, -2697825744960, 4744779429216, -8110465650176, 13523760003648, -22063059606912, 35250721087168, -55204237463376, 84944161233120, -128716413627520, 192127452262272, -282507110257440, 409984511707776]]
[25, [1, -50, 1200, -18400, 202450, -1702560, 11389600, -62339200, 285385200, -1114064850, 3779134240, -11360877600, 30885301600, -77374394400, 181284196800, -401031812480, 842726826450, -1691534395200, 3262594952400, -6077725252000, 10968671291040, -19220947905600, 32802221530400, -54693728246400, 89267475949600, -142743029326162, 223991253535200, -345718693238400, 525491835433600, -786742987874400, 1161295395893440]]
[26, [1, -52, 1300, -20800, 239252, -2107560, 14797120, -85155200, 410468500, -1688240372, 6030043240, -19043826880, 54177864000, -141373532200, 343594659200, -786344740480, 1706685733012, -3532307607400, 7010460268500, -13412537087040, 24832297608040, -44603133286400, 77924078814400, -132816314883200, 221394909419840, -361407558819164, 578472382307304, -909636568464000, 1407681394953600, -2145304809169704, 3221568346322560]]
[27, [1, -54, 1404, -23400, 280854, -2586168, 19014840, -114791040, 581257404, -2513392494, 9436118328, -31274226360, 93086496360, -253106219448, 638406025200, -1511651306880, 3387901252374, -7229247074928, 14765641285044, -29018207233080, 55108831380408, -101440703660400, 181434042582840, -316176906314880, 538265644820280, -896820392178414, 1464212332875816, -2346272971038368, 3696677780664960, -5733430059474216, 8759016189382896]]
[28, [1, -56, 1512, -26208, 327656, -3147984, 24189984, -152867520, 811401192, -3681079640, 14500933104, -50376047904, 156797510688, -444306558864, 1163495873088, -2851049839680, 6597606440936, -14512424533488, 30505974273096, -61591664700384, 119983597365744, -226303038736128, 414352982942304, -738358447579200, 1283861585339424, -2183013918429512, 3635198568784944, -5936645280445376, 9523977599887936, -15031150280922384, 23358459198626496]]

2018年9月23日日曜日

180923

Ruby


合同数157

D. Zagier により見つけられた例を確認してみる。

a = 411340519227716149383203r / 21666555693714761309610
b = 6803298487826435051217540r / 411340519227716149383203
c = 224403517704336969924557513090674863160948472041r / 8912332268928859588025535178967163570016480830
p d = a * a + b * b
p e = c * c
p d == e
p a * b / 2

出力結果
(50356938758080675904478428415148993121355253942510969278703974330010718396658421418332558705681/79429666471790634382465107301223380530614462469737965026693774053022299696758997817757488900)
(50356938758080675904478428415148993121355253942510969278703974330010718396658421418332558705681/79429666471790634382465107301223380530614462469737965026693774053022299696758997817757488900)
true
(157/1)

2018年9月17日月曜日

180917

Ruby


The sphere packing problem in dimension 8

Maryna Viazovska氏による講演が以下にアップロードされている。

https://www.youtube.com/watch?v=GSMoIb-D4p4
https://www.youtube.com/watch?v=0a2vdHlbJKc

前者の動画に出てくる
φ = 518400q + 31104000q^2 + 870912000q^3 + ... ,
ψ = q^(-1) + 144 - 5120q^(1/2) + 70524q - 626688q^(3/2) + ...
(ただし、q = exp(2π i t))
の係数を出力してみた。

require 'prime'

def power0(a, n)
  return 1 if n == 0
  k = power0(a, n >> 1)
  k *= k
  return k if n & 1 == 0
  return k * a
end

# x > 0
def sigma(x, i)
  sum = 1
  pq = i.prime_division
  pq.each{|a, n| sum *= (power0(a, (n + 1) * x) - 1) / (power0(a, x) - 1)}
  sum
end

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    ary << a[0] # Bn = a[0]
  }
  ary
end

def E_2k(k, n)
  a = -4 * k / bernoulli(2 * k)[-1]
  b = a.denominator
  c = a.numerator
  [b] + (1..n).map{|i| c * sigma(2 * k - 1, i)}
end

# m次以下を取り出す
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end

# m次以下を取り出す
def power(ary, n, m)
  return [1] if n == 0
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end

def I(ary, n)
  a = [1]
  i = 0
  while i < n
    a << -(0..i).inject(0){|s, j| s + ary[1 + i - j] * a[j]}
    i += 1
  end
  a
end

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end

def A000594(n)
  ary = [1]
  a = [0] + (1..n - 1).map{|i| s(i)}
  (1..n - 1).each{|i| ary << (1..i).inject(0){|s, j| s - 24 * a[j] * ary[-j]} / i}
  ary
end

def A(n)
  a = E_2k(1, n)
  b = E_2k(2, n)
  c = E_2k(3, n)
  d = mul(a, b, n)
  e = (1..n).map{|i| d[i] - c[i]}
  r = A000594(n)
  mul(mul(e, e, n - 1), I(r, n - 1), n - 1)
end

def A00_4(n)
  ary = [1] + [0] * n
  (1..n).each{|i|
    j = i * i
    break if j > n
    ary[j] = 2
  }
  power(ary, 4, n)
end

def A01_4(n)
  ary = [1] + [0] * n
  (1..n).each{|i|
    j = i * i
    break if j > n
    ary[j] = (-1) ** (i % 2) * 2
  }
  power(ary, 4, n)
end

def A10_4(n)
  ary = [1] + [0] * n
  (1..n).each{|i|
    j = i * (i + 1)
    break if j > n
    ary[j] = 1
  }
  power(ary, 4, n)
end

# -2~n-2 
def A319294(n)
  a = A00_4(n)
  b = A01_4(n)
  c = A10_4(n)
  d = (0..n).map{|i| a[i] + b[i]}
  e = I(mul(c, c, n), n)
  f = mul(d, e, n).map{|i| i / 2}
  g = [1] + (1..n).map{|i| b[i] - 16 * c[i - 1]}
  h = mul(g, I(mul(a, a, n), n), n).map{|i| i * 128}
  ([1, 0] + (2..n).map{|i| f[i] + h[i - 2]})[0..n]
end

n = 50
p a = A(n)
p A319294(n + 2)

# A281373
p a.map{|i| i / 518400}

出力結果
[518400, 31104000, 870912000, 15697152000, 210303475200, 2265242112000, 20595617280000, 163319763456000, 1156167168134400, 7434092726323200, 43996666201804800, 242175877622784000, 1250286900875712000, 6095989790740684800, 28230535439298662400, 124774907252150476800, 528508060756732224000, 2152911394014850944000, 8460229761257773363200, 32157626555717287833600, 118510579413170806579200, 424336487248949755392000, 1478950905909909993984000, 5025889410756351414681600, 16677888376529592125280000, 54116367018561184100198400, 171914123063311861284864000, 535276309312266116216832000, 1635212754461027241135244800, 4905795736572272712782438400, 14466381382727713044935884800, 41963590052233149723052032000, 119830807528853675615488512000, 337089191768542107424905984000, 934712156683604751871271731200, 2556376529361173779444614604800, 6899624930489021508910336704000, 18386710788968496778380304896000, 48402789412667653557030560256000, 125927725910609020521612819456000, 323923060378851737609682025228800, 824149276407205470840652886016000, 2074805192196777843429565441536000, 5170241957113295217494592168960000, 12757119890783853676424057997350400, 31177470401176483400583413586124800, 75492971663046873095674098385920000, 181164766163746719677990358085632000, 430983257182115546274947305704403200, 1016666528235478923452749998848640000]
[1, 0, 144, -5120, 70524, -626688, 4265600, -24164352, 119375370, -529539072, 2151757440, -8125793280, 28827864296, -96885780480, 310514729472, -954123868160, 2823202073655, -8074060259328, 22387521828480, -60344692402176, 158484892943628, -406368240128000, 1019049374174976, -2503142549913600, 6030967901500134, -14270261339381760, 33196945984012800, -76000499056847872, 171386626286516040, -381008035914817536, 835625106694349824, -1809276617175275520, 3869780788718208819, -8180990261004369920, 17103874636380714624, -35380669867248697344, 72446742810305722680, -146905267940338876416, 295116441246442028160, -587553474790036682752, 1159710012298896697386, -2270074082571419443200, 4408097952551024955136, -8493898810748820572160, 16245174113200620330624, -30847020761299021213696, 58167128640245120017920, -108947149043693266262016, 202731394027206007021980, -374870460273855469240320, 688938058997189908185600, -1258627893976024198476800, 2286168896136927086572356]
[1, 60, 1680, 30280, 405678, 4369680, 39729200, 315045840, 2230260741, 14340456648, 84870112272, 467160257760, 2411818867430, 11759239565472, 54457051387536, 240692336520352, 1019498573990610, 4152992658207660, 16319887656747248, 62032458633713904, 228608370781579488, 818550322625288880, 2852914556153375760, 9695002721366418624, 32171852578182083575, 104391140082101049576, 331624465785709608960, 1032554609012859020480, 3154345591167105017622, 9463340541227377918176, 27905828284582779793472, 80948283279770736348480, 231155107115844281665680, 650249212516477830680760, 1803071289898928919504768, 4931281885341770407879272, 13309461671468019886015310, 35468192108349723723727440, 93369578342337294670197840, 242916137944847647611135840, 624851582520933135821145882, 1589794128871924133566074240, 4002324830626500469578637040, 9973460565419165157204074400, 24608641764629347369645173606, 60141725310911426312853807072, 145626874350013258286408368800, 349469070531918826539333252480, 831372023885253754388401438473, 1961162284404859034438175152100]

2018年9月9日日曜日

180909

Ruby


1^(1^m) * 2^(2^m) * ... * n^(n^m) の漸近展開(3)

m が奇数のとき、数列{a_m(k)} を出力してみた。

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    ary << a[0] # Bn = a[0]
  }
  ary
end

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

# k は奇数
def A(k, n)
  a = bernoulli(2 * n + k + 1)
  ary = [1]
  (1..n).each{|i|
    ary << -1 / (2r * i) * (0..i - 1).inject(0){|s, j| s + a[2 * i - 2 * j + k + 1] * ary[j] / (ncr(2 * i - 2 * j + k, k) * (2 * i - 2 * j + k + 1r))}
  }
  ary
end

n = 15
(0..10).each{|i| p [2 * i + 1, A(2 * i + 1, n)]}

出力結果
[1, [1, (1/720), (-1433/7257600), (1550887/15676416000), (-365236274341/3476402012160000), (31170363588856607/162695614169088000000), (-2626723351027654662151/4919915372473221120000000), (127061942835077684151157039/60219764159072226508800000000), (-5696145248370283185291966600124423/507464726196802564122476544000000000), (254326794362835881966596504823903633657/3288371425755280615513648005120000000000), (-33203124408022060010631772664020406983485604379/49554442037561776763544469977956352000000000000), (5190609778186715223789005366945445257148700136639/728146903409071005505143232329154560000000000000), (-2168800830257199852481630419559118897078575013107378361693/23736657023099351168580622790593057338163200000000000000), (688254812126920339705848920425635906853150484332853559547459091/495621398642314452399963403867583037220847616000000000000000), (-455094637798219508780751141299232276619401642948330067032553732024790741/18429741183082299990226927168424539349021685102673920000000000000000), (1314225236181088962408003668993914937576664551129347651283560067977220345649007/2587535662104754918627860574446805324602644588415418368000000000000000000)]]
[3, [1, (-1/5040), (1513/50803200), (-127057907/8449588224000), (7078687551763/442893616349184000), (-1626209947417109183/55804595659997184000000), (25620826938516570309695021/315568291905804875857920000000), (-67861652779316417663427293866727/211531737430299124385080934400000000), (11129902336987204608540488473560076627/6522145617145034649275530739712000000000), (-2992048697379116617363098289271338606184087563/254485460571619683408716971558739902464000000000), (593799837691907572156765292649932318031816367209421/5830030551277105474454243348436586856448000000000000), (-350228022676002451254790765950432151754187682917945416221/323216893762802727503743251237324375321477120000000000000), (19139693470931372486110838612190067457694071143730249104897863134207/1378125572143854007330601197963918331092507560404582400000000000000), (-45448299271539740554434251503469985665183859239384209464213143704179321221/215318339391755750105333131169882600049893381237611954176000000000000000), (1397991844435921714223037335622050696152582575927095990021849624899727979354737/372470165835049585575759753511730850233405242093409148705177600000000000000), (-18531750464382297398610536324979225903769771871072938032216954843645084341184560943687277/240053297179031107407721403540775415666927344476781262248999911424000000000000000000)]]
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2018年9月2日日曜日

180902(3)

Ruby


1^(1^m) * 2^(2^m) * ... * n^(n^m) の漸近展開(2)

数列{a_m(k)} の分子を出力してみた。

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    ary << a[0] # Bn = a[0]
  }
  ary
end

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

def A(k, n)
  a = bernoulli(n + k + 1)
  ary = [1]
  (1..n).each{|i|
    ary << (-1) ** (k % 2) / i.to_r * (0..i - 1).inject(0){|s, j| s + a[i - j + k + 1] * ary[j] / (ncr(i - j + k, k) * (i - j + k + 1r))}
  }
  ary
end

n = 15
(0..10).each{|i| p [i, A(i, n).map{|i| i.numerator}]}

出力結果
[0, [1, 1, 1, -139, -571, 163879, 5246819, -534703531, -4483131259, 432261921612371, 6232523202521089, -25834629665134204969, -1579029138854919086429, 746590869962651602203151, 1511513601028097903631961, -8849272268392873147705987190261]]
[1, [1, 0, 1, 0, -1433, 0, 1550887, 0, -365236274341, 0, 31170363588856607, 0, -2626723351027654662151, 0, 127061942835077684151157039, 0]]
[2, [1, -1, 1, 259193, -1036793, -201551328007, 9137074752049, 9142431862033871923, -11105299580705049589, -11003865617473929216508154207, 114467620015003245418244743007, 32505236416490926096399421788847363, -254505521478572052318535393350091231, -1828472168539763642032546635313363411876021, 449843951207507728031832088068767125946389963, 31024838101634458799200044021590825216299304842344313]]
[3, [1, 0, -1, 0, 1513, 0, -127057907, 0, 7078687551763, 0, -1626209947417109183, 0, 25620826938516570309695021, 0, -67861652779316417663427293866727, 0]]
[4, [1, 1, 1, -476279, -1905119, 50409422809211, 327409436642411, -33180456307810635797257, -282922303030975330914097, 5379031277933782422359791052783, 47439700106584179998732858654521, -36404043230640972806023058818270566601769, -448477438414484751987416795754334219171169, 44059016262452855253179633655261374898258492303313, 8158862019359543444070175550192616738556343315215309, -569629821149561571872765829220003512513888474356694799393913]]
[5, [1, 0, 1, 0, -33589, 0, 6685358543, 0, -3154174722920027, 0, 4928758950963155306741, 0, -2256715986149278985600498968703, 0, 629222853795800878962230085722916097, 0]]
[6, [1, -1, 1, 9407989, -37631989, -2096938112640143, 149904329568001573, 2077435474826915904958427, -17724254999346713040382427, -111976939829427960973528888190502151, 15178929891207525282519637681009903963, 1779598085747305754736646575789869852177394973, -241225547842987701588833000525699972846584624703, -2216804467382185051458528121403529509855199450770516869, 1857837620155651450279891034916557225570091149066824757, 19922345484131707240270922592637479980113734425892164090660921569859]]
[7, [1, 0, -1, 0, 6569539, 0, -94862020867, 0, 148360361562210426521, 0, -3758214620551052982685723, 0, 17659775228092034903641523829434461, 0, -15253807614781285310910350330441645179853, 0]]
[8, [1, 1, 1, -975236429, -3900952541, 1673187234045287, 24741204553616603957, -416390509638217044924161981, -3553044445676273323476205229, 2594326466683133286611419169781551183899, 318285642111072314088164648150484807919, -183637667316299523050446787340969584371799407441, -5148483739079534617735950371518901403202578056384131, 28638181650561498070967147583289414129520178187169628122173, 18827705277362052458863395307577239815778846623862046296474289, -4107665650991246957700860278306395520803885006941136705235066280532998941]]
[9, [1, 0, 691, 0, -756278519, 0, 70428516045253307, 0, -1207282026600970337462141, 0, 674487004522944662914168709557673, 0, -15888760614678571271038978643189988077091187, 0, 1583798222362424121078560062946355762334601480141783, 0]]
[10, [1, -691, 477481, 27270129276629, -75375321284918639, -365968280308506388285675667, 1643735859311321887269714365897, 28818902191146659899637620909080751658287, -169930095546261471299784052384223123018804317, -382217632900099668663243861277991747464040649775752953, 46823037076414380661715849077881931738040803765228528402891, 4866716392353709570566815278008611320497715893298702739601120942213337, -41443567019975441388846205026673443833189485265378666543917124727811255867, -329646173828465774346971211550445315710876341492464017680118089036106858298781587903, 61664000792126295378754033137652704976705049804896776894833298530688749725982457644122487, 13386120102253131524884309722845350152952297269142174938072503858518945291821013177207927479272809211]]

180902(2)

Ruby


1^(1^m) * 2^(2^m) * ... * n^(n^m) の漸近展開(1)

m = 0 のとき
1 * 2 * ... * n ~ √(2 * π) n^(n + 1/2) * e^(-n) * (1 + 1/(12 * n) + ... )
となるが、
m が1以上のとき、
generalized Glaisher-Kinkelin constant A_m、
適当な多項式U(n), V(n)
および
数列{a_m(k)}
を用いて
1^(1^m) * 2^(2^m) * ... * n^(n^m) ~ A_m * n^U(n) * e^V(n) * Product_{k>=0} a_m(k)/n^k
と書ける。
数列{a_m(k)} を出力してみた。

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    ary << a[0] # Bn = a[0]
  }
  ary
end

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

def A(k, n)
  a = bernoulli(n + k + 1)
  ary = [1]
  (1..n).each{|i|
    ary << (-1) ** (k % 2) / i.to_r * (0..i - 1).inject(0){|s, j| s + a[i - j + k + 1] * ary[j] / (ncr(i - j + k, k) * (i - j + k + 1r))}
  }
  ary
end

n = 15
(0..10).each{|i| p [i, A(i, n)]}

出力結果
[0, [1, (1/12), (1/288), (-139/51840), (-571/2488320), (163879/209018880), (5246819/75246796800), (-534703531/902961561600), (-4483131259/86684309913600), (432261921612371/514904800886784000), (6232523202521089/86504006548979712000), (-25834629665134204969/13494625021640835072000), (-1579029138854919086429/9716130015581401251840000), (746590869962651602203151/116593560186976815022080000), (1511513601028097903631961/2798245444487443560529920000), (-8849272268392873147705987190261/299692087104605205332754432000000)]]
[1, [1, (0/1), (1/720), (0/1), (-1433/7257600), (0/1), (1550887/15676416000), (0/1), (-365236274341/3476402012160000), (0/1), (31170363588856607/162695614169088000000), (0/1), (-2626723351027654662151/4919915372473221120000000), (0/1), (127061942835077684151157039/60219764159072226508800000000), (0/1)]]
[2, [1, (-1/360), (1/259200), (259193/1959552000), (-1036793/2821754880000), (-201551328007/5079158784000000), (9137074752049/76796880814080000000), (9142431862033871923/304115648023756800000000), (-11105299580705049589/125121866615488512000000000), (-11003865617473929216508154207/258236518070374430146560000000000), (114467620015003245418244743007/929651465053347948527616000000000000), (32505236416490926096399421788847363/334674527419205261469941760000000000000), (-254505521478572052318535393350091231/920050700832433373350094438400000000000000), (-1828472168539763642032546635313363411876021/5630710289094492244902577963008000000000000000), (449843951207507728031832088068767125946389963/490547480385912164375912592137256960000000000000000), (31024838101634458799200044021590825216299304842344313/20724188259597771556163554333704585216000000000000000000)]]
[3, [1, (0/1), (-1/5040), (0/1), (1513/50803200), (0/1), (-127057907/8449588224000), (0/1), (7078687551763/442893616349184000), (0/1), (-1626209947417109183/55804595659997184000000), (0/1), (25620826938516570309695021/315568291905804875857920000000), (0/1), (-67861652779316417663427293866727/211531737430299124385080934400000000), (0/1)]]
[4, [1, (1/1260), (1/3175200), (-476279/12002256000), (-1905119/60491370240000), (50409422809211/4192051957632000000), (327409436642411/31691912799697920000000), (-33180456307810635797257/3633794721613363507200000000), (-282922303030975330914097/36628650793862704152576000000000), (5379031277933782422359791052783/415368900002403065090211840000000000), (47439700106584179998732858654521/4428471503102543447807950848000000000000), (-36404043230640972806023058818270566601769/1233152174753934248476601993134080000000000000), (-448477438414484751987416795754334219171169/18645260882279485836966222136187289600000000000000), (44059016262452855253179633655261374898258492303313/446367545521770890936971357940323713024000000000000000), (8158862019359543444070175550192616738556343315215309/102361005539052500709666271802875033870663680000000000000000), (-569629821149561571872765829220003512513888474356694799393913/1251814885386412641031742229871630561277116416000000000000000000)]]
[5, [1, (0/1), (1/10080), (0/1), (-33589/2235340800), (0/1), (6685358543/878757175296000), (0/1), (-3154174722920027/389746382387281920000), (0/1), (4928758950963155306741/333934700429423149056000000), (0/1), (-2256715986149278985600498968703/54873539142916598253181992960000000), (0/1), (629222853795800878962230085722916097/3871876921924195172744521423257600000000), (0/1)]]
[6, [1, (-1/1680), (1/5644800), (9407989/312947712000), (-37631989/2103008624640000), (-2096938112640143/229648541810688000000), (149904329568001573/25463430315969085440000000), (2077435474826915904958427/299449940515796444774400000000), (-17724254999346713040382427/4024607200532304217767936000000000), (-111976939829427960973528888190502151/11379335383073058237485772963840000000000), (15178929891207525282519637681009903963/2485246847663155919066892815302656000000000000), (1779598085747305754736646575789869852177394973/79329079377407936936615218664460779520000000000000), (-241225547842987701588833000525699972846584624703/17592016642733984095063790891030822466355200000000000000), (-2216804467382185051458528121403529509855199450770516869/29554587959793093279707168696931781743476736000000000000000), (1857837620155651450279891034916557225570091149066824757/40889641694960797290512506338343265094504284160000000000000000), (19922345484131707240270922592637479980113734425892164090660921569859/57613815909476645084391529325773831926971254613999616000000000000000000)]]
[7, [1, (0/1), (-1/9504), (0/1), (6569539/410983372800), (0/1), (-94862020867/11717957925273600), (0/1), (148360361562210426521/17228547937242505543680000), (0/1), (-3758214620551052982685723/239312482485807898542735360000), (0/1), (17659775228092034903641523829434461/403596864162581236612265335062528000000), (0/1), (-15253807614781285310910350330441645179853/88223045731026957673548304121988120576000000), (0/1)]]
[8, [1, (1/1188), (1/2822688), (-975236429/22886636572800), (-3900952541/108757296993945600), (1673187234045287/129203668828807372800), (24741204553616603957/2095192534461706118799360000), (-416390509638217044924161981/42314508425988616775271874560000), (-3553044445676273323476205229/402157088080595813832183895818240000), (2594326466683133286611419169781551183899/185861603494377898333565624003980689408000000), (318285642111072314088164648150484807919/25976892347214228614150113096085771649024000000), (-183637667316299523050446787340969584371799407441/5770922496287724172005132524974030686460575744000000), (-5148483739079534617735950371518901403202578056384131/187164866768601985436139260102967752835563976759705600000000), (28638181650561498070967147583289414129520178187169628122173/269162779978172665792477323318604783077839478999062937600000000), (18827705277362052458863395307577239815778846623862046296474289/206568437168688656017105136826216603563521752478872853335244800000000), (-4107665650991246957700860278306395520803885006941136705235066280532998941/8374387727037222459261450799503234216766953606369744910637491814400000000000)]]
[9, [1, (0/1), (691/3603600), (0/1), (-756278519/25971865920000), (0/1), (70428516045253307/4773203017494912000000), (0/1), (-1207282026600970337462141/76897311407776148889600000000), (0/1), (674487004522944662914168709557673/23554107868070281061777817600000000000), (0/1), (-15888760614678571271038978643189988077091187/199127501983984940101086087058882560000000000000), (0/1), (1583798222362424121078560062946355762334601480141783/5023031063046416911037916763277724352512000000000000000), (0/1)]]
[10, [1, (-691/360360), (477481/259718659200), (27270129276629/280776648087936000), (-75375321284918639/404722691619874467840000), (-365968280308506388285675667/12396898877931726874619904000000), (1643735859311321887269714365897/26804078877908862579228171632640000000), (28818902191146659899637620909080751658287/1284662675970950616633738302628574003200000000), (-169930095546261471299784052384223123018804317/3703528335303134113681071477881863422345216000000000), (-382217632900099668663243861277991747464040649775752953/12011431238188536682854998259925574725886898339840000000000), (46823037076414380661715849077881931738040803765228528402891/735834691368915583435716619400952618397502456576606208000000000000), (4866716392353709570566815278008611320497715893298702739601120942213337/67086843513570712170664394764733803248128168268742290718064640000000000000), (-41443567019975441388846205026673443833189485265378666543917124727811255867/290104979142604102053847455569033680061945600607887662597941284044800000000000000), (-329646173828465774346971211550445315710876341492464017680118089036106858298781587903/1359048993689774584809618098155140700312595316255759175219323574538993664000000000000000), (61664000792126295378754033137652704976705049804896776894833298530688749725982457644122487/130272674167368547055610398108415609735396061612136150117621427923351887297904640000000000000000), (13386120102253131524884309722845350152952297269142174938072503858518945291821013177207927479272809211/11970990520052997052324739580898905526683067814450092679378444730447066957201593597952000000000000000000)]]

180902

Ruby


Expansion of ((1 + k*x)/(1 - k*x))^(m/k)(5)

k = 1 のときを出力してみた。

def A(k, m, n)
  a, b = 1, 2 * m
  ary = [1]
  cnt = 0
  while cnt < n
    a, b = b, (2 * m * b + k * k * cnt * a) / (cnt + 2r)
    a = a.to_i if a.denominator == 1
    ary << a
    cnt += 1
  end
  ary
end

n = 25
(1..100).each{|i| p [i, A(1, i, n)]}

出力結果
[1, [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]]
[2, [1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100]]
[3, [1, 6, 18, 38, 66, 102, 146, 198, 258, 326, 402, 486, 578, 678, 786, 902, 1026, 1158, 1298, 1446, 1602, 1766, 1938, 2118, 2306, 2502]]
[4, [1, 8, 32, 88, 192, 360, 608, 952, 1408, 1992, 2720, 3608, 4672, 5928, 7392, 9080, 11008, 13192, 15648, 18392, 21440, 24808, 28512, 32568, 36992, 41800]]
[5, [1, 10, 50, 170, 450, 1002, 1970, 3530, 5890, 9290, 14002, 20330, 28610, 39210, 52530, 69002, 89090, 113290, 142130, 176170, 216002, 262250, 315570, 376650, 446210, 525002]]
[6, [1, 12, 72, 292, 912, 2364, 5336, 10836, 20256, 35436, 58728, 93060, 142000, 209820, 301560, 423092, 581184, 783564, 1038984, 1357284, 1749456, 2227708, 2805528, 3497748, 4320608, 5291820]]
[7, [1, 14, 98, 462, 1666, 4942, 12642, 28814, 59906, 115598, 209762, 361550, 596610, 948430, 1459810, 2184462, 3188738, 4553486, 6376034, 8772302, 11879042, 15856206, 20889442, 27192718, 35011074, 44623502]]
[8, [1, 16, 128, 688, 2816, 9424, 27008, 68464, 157184, 332688, 658048, 1229360, 2187520, 3732560, 6140800, 9785072, 15158272, 22900496, 33830016, 48978352, 69629696, 97364944, 134110592, 182192752, 244396544, 324031120]]
[9, [1, 18, 162, 978, 4482, 16722, 53154, 148626, 374274, 864146, 1854882, 3742290, 7159170, 13079250, 22952610, 38878482, 63821826, 101880594, 158611106, 241419474, 360027522, 527022162, 758497698, 1074801042, 1501390338, 2069818002]]
[10, [1, 20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, 4780008, 10377180, 21278640, 41517060, 77548920, 139380012, 242080320, 407782740, 668274440, 1068305020, 1669752016, 2556801700, 3842321560, 5675620300, 8251811680, 11823020020]]
[11, [1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898, 26572086, 58227906, 121023606, 240089586, 457018518, 838478850, 1488341910, 2564399090, 4300978550, 7039035586, 11265589302, 17664712562, 27182654422, 41110086402, 61184918102]]
[12, [1, 24, 288, 2312, 14016, 68664, 284000, 1022760, 3281280, 9545560, 25534368, 63521352, 148321344, 327572856, 688686048, 1385794152, 2681291520, 5008112280, 9060853280, 15926230920, 27266245056, 45570869944, 74501171808, 119348538792, 187641279616, 289936284120]]
[13, [1, 26, 338, 2938, 19266, 101946, 454610, 1761370, 6065410, 18892250, 53972178, 143027898, 354870594, 830764794, 1847023698, 3921503898, 7988589570, 15677993370, 29746958930, 54734043130, 97926519106, 170763634106, 290835675858, 484685386458, 791675204866, 1269252768602]]
[14, [1, 28, 392, 3668, 25872, 147084, 703640, 2919620, 10746400, 35704060, 108568488, 305568564, 803467056, 1989102444, 4666890936, 10435418532, 22345512000, 46012094940, 91437047240, 175918049300, 328578611536, 597268764748, 1058868074712, 1834389137028, 3110749728352, 5171677701820]]
[15, [1, 30, 450, 4510, 34050, 207006, 1057730, 4680990, 18347010, 64797470, 209070018, 623207070, 1732242690, 4524812190, 11180805570, 26283115038, 59064045570, 127421652510, 264870794690, 532225891230, 1036722552066, 1962569928350, 3618706767810, 6511963979550, 11457102844930, 19739530275102]]
[16, [1, 32, 512, 5472, 44032, 285088, 1549824, 7288544, 30316544, 113461024, 387328512, 1219605600, 3575055360, 9832110240, 25537728000, 63001648608, 148348809216, 334834507296, 727126954496, 1524223640416, 3093172083712, 6092464564128, 11673741260288, 21804412007648, 39773478832128, 70970111952160]]
[17, [1, 34, 578, 6562, 56066, 385186, 2220098, 11058466, 48663554, 192441122, 693230658, 2300164770, 7094825730, 20501991330, 55871829570, 144411206178, 355761664002, 838944980514, 1900906442306, 4152257037218, 8769652761346, 17955289409186, 35721495233602, 69199648501538, 130777539341314, 241521130125602]]
[18, [1, 36, 648, 7788, 70416, 511668, 3116952, 16395516, 76117536, 317222212, 1202893992, 4196289420, 13591279920, 41188096980, 117561917880, 317844953628, 818017823808, 2012724468324, 4752575891144, 10805739370668, 23727649169232, 50452591339764, 104129375982552, 209050519717692, 409027707560544, 781326377027460]]
[19, [1, 38, 722, 9158, 87362, 669446, 4298066, 23810534, 116323586, 509663334, 2029779538, 7428962950, 25216532290, 79995909190, 238745924050, 674152795558, 1810015572994, 4640757865126, 11406058224594, 26964373486406, 61497762026306, 135678002535302, 290259969857618, 603439865557862, 1221518092836098, 2411872177424102]]
[20, [1, 40, 800, 10680, 107200, 864008, 5831520, 33940120, 174074240, 800061160, 3339504032, 12798246520, 45443741760, 150656183240, 469398016480, 1382296736088, 3866465104640, 10317238542760, 26364054632480, 64734486343480, 153196621856192, 350372386417800, 776310358810720, 1670010194226200, 3494968152620160, 7128358422880360]]
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