180423

Ruby


Expansion of Product_{k>=1} ((1 + b*x^k)/(1 - b*x^k))^(1/b)

b が32以下のとき、出力してみた。

def s(k, n)
  s = 0
  (1..n).each{|i| s += (n / i) * k ** (i - 1) * (1 - (-1) ** (i % 2)) if n % i == 0}
  s
end

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

n = 32
(1..n).each{|i|
  p i
  ary = A(i, n)
  ary = ary.map(&:to_i) if ary.all?{|i| i.denominator == 1}
  p ary
}

出力結果
1
[0, 2, 4, 8, 8, 12, 16, 16, 16, 26, 24, 24, 32, 28, 32, 48, 32, 36, 52, 40, 48, 64, 48, 48, 64, 62, 56, 80, 64, 60, 96, 64, 64]
[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, 150144, 192612]
2
[0, 2, 4, 14, 8, 42, 28, 142, 16, 554, 84, 2070, 56, 8218, 284, 32934, 32, 131106, 1108, 524326, 168, 2097634, 4140, 8388654, 112, 33554642, 16436, 134219390, 568, 536870970, 65868, 2147483710, 64]
[1, 2, 4, 10, 18, 38, 72, 142, 260, 510, 940, 1814, 3362, 6490, 12112, 23466, 44114, 85766, 162516, 317190, 604806, 1184682, 2271248, 4461514, 8591784, 16916490, 32696708, 64496130, 125037142, 247007142, 480077432, 949510526, 1849375796]
3
[0, 2, 4, 24, 8, 172, 48, 1472, 16, 13194, 344, 118120, 96, 1062908, 2944, 9566544, 32, 86093476, 26388, 774841016, 688, 6973573344, 236240, 62762119264, 192, 564859073822, 2125816, 5083731696240, 5888, 45753584909980, 19133088, 411782264189360, 64]
[1, (2/1), (4/1), (40/3), (74/3), (232/3), (1496/9), (4624/9), (10340/9), (296618/81), (700552/81), (2238824/81), (16426040/243), (52645480/243), (132141152/243), (1269387136/729), (3247054622/729), (10377298480/729), (242469841436/6561), (773105011720/6561), (2030882001800/6561), (19381160551952/19683), (51408848071432/19683), (163173266108720/19683), (1309070336102008/59049), (4146469657789322/59049), (11165827512192184/59049), (953107863703822048/1594323), (2582077601954622320/1594323), (8148785562277449496/1594323), (66578169249950838928/4782969), (209773435555040155856/4782969), (573976203565114384516/4782969)]
4
[0, 2, 4, 38, 8, 522, 76, 8206, 16, 131186, 1044, 2097174, 152, 33554458, 16412, 536872638, 32, 8589934626, 262372, 137438953510, 2088, 2199023280394, 4194348, 35184372088878, 304, 562949953423922, 67108916, 9007199255134550, 32824, 144115188075855930, 1073745276, 2305843009213694014, 64]
[1, 2, 4, 18, 34, 166, 384, 1902, 4756, 24022, 64284, 321542, 899658, 4455690, 12888944, 63185250, 187513426, 910880550, 2759413788, 13295839638, 40967821494, 195979968882, 612569599440, 2911592648458, 9213101043936, 43538337410474, 139246245625364, 654607535452762, 2113349714232038, 9887935161585494, 32189873102090592, 149956946909271358, 491842446413404948]
5
[0, 2, 4, 56, 8, 1260, 112, 31264, 16, 781418, 2520, 19531272, 224, 488281276, 62528, 12207035280, 32, 305175781284, 1562836, 7629394531288, 5040, 190734863375392, 39062544, 4768371582031296, 448, 119209289550787550, 976562552, 2980232238771875504, 125056, 74505805969238281308, 24414070560, 1862645149230957031312, 64]
[1, (2/1), (4/1), (24/1), (46/1), (1688/5), (4056/5), (30192/5), (79892/5), (595218/5), (8482312/25), (61950664/25), (185810568/25), (1333303928/25), (4156712512/25), (146798821632/125), (471794234774/125), (3287111258608/125), (10829625801228/125), (74554916419912/125), (1253843771372744/625), (8541147357705648/625), (29240326213347256/625), (197316578237348016/625), (685913250477493464/625), (114739470095081542282/15625), (404242361688650228664/15625), (2684200248207346016928/15625), (9569887602474737888368/15625), (63102127188981411612872/15625), (1136908484423542678089936/78125), (7448679553814096816675632/78125), (27099225934636021162314324/78125)]
6
[0, 2, 4, 78, 8, 2602, 156, 93326, 16, 3359466, 5204, 120932374, 312, 4353564698, 186652, 156728336358, 32, 5642219814946, 6718932, 203119913336870, 10408, 7312316880406434, 241864748, 263243407684534318, 624, 9476762676643246802, 8707129396, 341163456359166494910, 373304, 12281884428929630994490, 313456672716, 442147839441466715799614, 64]
[1, (2/1), (4/1), (94/3), (182/3), (1906/3), (14024/9), (147454/9), (402596/9), (37594526/81), (111039724/81), (1122328694/81), (10511211398/243), (103896051694/243), (338373914096/243), (9842085183322/729), (33151290370946/729), (316116032309014/729), (9851627050052372/6561), (92585416237240390/6561), (328142293519928774/6561), (9133556772972780542/19683), (33023258763732722224/19683), (302897450100868035614/19683), (3342995873851762827304/59049), (30347640997363316380826/59049), (113367633670608538017700/59049), (27527686846183407773527654/1594323), (104244376755075845040850274/1594323), (929499020598599117068883794/1594323), (10689854077727858017543383160/4782969), (94569068435206130047770890894/4782969), (366578178242754231255864512404/4782969)]
7
[0, 2, 4, 104, 8, 4812, 208, 235312, 16, 11529914, 9624, 564950520, 416, 27682574428, 470624, 1356446160624, 32, 66465861139236, 23059828, 3256827195820936, 19248, 159584532595930624, 1129901040, 7819642097165976144, 832, 383162462761132852862, 55365148856, 18774960675295543201040, 941248, 919973073089479921953660, 2712892321248, 45078680581384516175726560, 64]
[1, (2/1), (4/1), (40/1), (78/1), (1112/1), (2760/1), (273376/7), (760700/7), (10518934/7), (31761048/7), (426145032/7), (1364043816/7), (17844487720/7), (418161109872/49), (5353454555712/49), (18575746669614/49), (233426476330288/49), (834281030427084/49), (10314444861325624/49), (37797669650029160/49), (3224325449130404816/343), (12072631650081478968/343), (145238278293986308064/343), (554123099905409557752/343), (6589239632039780868566/343), (25560996349807351214536/343), (300756570493458196811360/343), (8289017173057792232444960/2401), (96591279742277864259510920/2401), (385422918708499664956340368/2401), (4451476268284753799145232480/2401), (17979513759932392301195898724/2401)]
8
[0, 2, 4, 134, 8, 8202, 268, 524302, 16, 33554834, 16404, 2147483670, 536, 137438953498, 1048604, 8796093047454, 32, 562949953421346, 67109668, 36028797018964006, 32808, 2305843009215267754, 4294967340, 147573952589676412974, 1072, 9444732965739290468402, 274877906996, 604462909807314688017590, 2097208, 38685626227668133590597690, 17592186094908, 2475880078570760549798248510, 64]
[1, 2, 4, 50, 98, 1830, 4576, 83950, 236500, 4211766, 12903260, 222377926, 723722602, 12136867530, 41382435824, 678060771778, 2400028798290, 38546050682278, 140724756748476, 2220907298526934, 8323586858891766, 129340015891714962, 495838256186203600, 7598518664789468426, 29712560805044620096, 449653168709476240810, 1789424007528983041364, 26772438897078377653946, 108230158789714892601894, 1602415158326270199908630, 6570474614516939785824512, 96345625313855031020927294, 400181545721010824307552148]
9
[0, 2, 4, 168, 8, 13132, 336, 1062896, 16, 86093946, 26264, 6973568824, 672, 564859072988, 2125792, 45753584950128, 32, 3706040377703716, 172187892, 300189270593998280, 52528, 24315330918117047424, 13947137648, 1969541804367222465808, 1344, 159532886153745019792382, 1129718145976, 12922163778453346856146320, 4251584, 1046695266054721074427023100, 91507169900256, 84782316550432407028588866464, 64]
[1, (2/1), (4/1), (184/3), (362/3), (8584/3), (64712/9), (1494880/9), (4247108/9), (853108250/81), (2640071080/81), (56917680248/81), (562170765704/243), (11776837529032/243), (40672751846672/243), (2494620913395520/729), (8953625447325278/729), (179248555792890544/729), (5978105024741472716/6561), (117498158146274411896/6561), (447374926352282750312/6561), (25952222443266127943696/19683), (101156320830959509321432/19683), (1927627631549274397267232/19683), (23008517384423901010298440/59049), (432687653842400921459638442/59049), (1753235643083340676831562296/59049), (879539538019851762208224113632/1594323), (3622612594381314456996141973280/1594323), (66568976375063388896174552784664/1594323), (834792764845975268282330904391696/4782969), (15184470400865451509706745060566176/4782969), (64332469411506587867134118181457348/4782969)]
10
[0, 2, 4, 206, 8, 20010, 412, 2000014, 16, 200000618, 40020, 20000000022, 824, 2000000000026, 4000028, 200000000061030, 32, 20000000000000034, 400001236, 2000000000000000038, 80040, 200000000000006001442, 40000000044, 20000000000000000000046, 1648, 2000000000000000000100050, 4000000000052, 200000000000000000600001854, 8000056, 20000000000000000000000000058, 400000000122060, 2000000000000000000000000000062, 64]
[1, (2/1), (4/1), (74/1), (146/1), (21438/5), (54056/5), (1535942/5), (4390292/5), (120110518/5), (1872052812/25), (49407669414/25), (164030172818/25), (4202245608778/25), (14648393845712/25), (1829654153874882/125), (6633662729547274/125), (162150393258821358/125), (607422942798629828/125), (14567331150184714862/125), (280533821134946563494/625), (6614982975852088461898/625), (26097759569345430506256/625), (606118849301109823463466/625), (2442295290503447959525864/625), (1398715653778159931368712282/15625), (5742664946531353806250416164/15625), (129917201568792788536391188178/15625), (542447976542524623973615203718/15625), (12131647919659571374608691764822/15625), (257154130067036836506421179246936/78125), (5690464085643148106563807777369382/78125), (24461209239959985809539218076748324/78125)]
11
[0, 2, 4, 248, 8, 29292, 496, 3543136, 16, 428718506, 58584, 51874849224, 992, 6276856753468, 7086272, 759499667255568, 32, 91899459727144356, 857437012, 11119834626984463000, 117168, 1345499989865130649504, 103749698448, 162805498773679522226688, 1984, 19699465351615222189570142, 12553713506936, 2383635307545441886206421040, 14172544, 288419872212998468075352128220, 1518999334511136, 34898804537772814637117607507664, 64]
[1, (2/1), (4/1), (88/1), (174/1), (6200/1), (15672/1), (537328/1), (1542788/1), (50800298/1), (159214280/1), (55577026712/11), (185665682264/11), (5714756455400/11), (1823589336320/1), (601670225687040/11), (2197995169333466/11), (64472898791113104/11), (22135181028668476/1), (7003777944049223096/11), (27208919907806598040/11), (769174498693684247440/11), (33684879069154367132344/121), (937515400504395602512880/121), (3813686634970391575368024/121), (9514039909377367941253246/11), (433947965618241299731230776/121), (11755915279134764410419673248/121), (49590897847083917648043192368/121), (120695443898678508243801070552/11), (5688390316669900527425338299280/121), (150635209895216495789821192588336/121), (654633235881544378068466325410820/121)]
12
[0, 2, 4, 294, 8, 41482, 588, 5971982, 16, 859964274, 82964, 123834728470, 1176, 17832200896538, 11943964, 2567836929223614, 32, 369768517790072866, 1719928548, 53246666561770487846, 165928, 7667519984894968162314, 247669456940, 1104122877824872835186734, 2352, 158993694406781688267090482, 35664401793076, 22895091994576563113011055190, 23887928, 3296893247219025087902087381050, 5135673858447228, 474752627599539612657900582862910, 64]
[1, (2/1), (4/1), (310/3), (614/3), (26098/3), (198272/9), (8072926/9), (23258420/9), (8169353654/81), (25709754844/81), (966265025510/81), (9730275102014/243), (354495349948030/243), (1250953722806096/243), (133169575019108626/729), (489318402319760258/729), (16972919850924597046/729), (580501613047581639836/6561), (19738094406100644756310/6561), (77189741422741353565814/6561), (7735454015070286298984726/19683), (31012275908690281338314608/19683), (1019594146311005789784399326/19683), (12533959594939036964560538848/59049), (406181997712348413331365433658/59049), (1697094728808469155952156639540/59049), (1465503324398447438369722158596398/1594323), (6231048084598918807559949112962962/1594323), (196891532700105504114637626136351714/1594323), (2551517098036808090576242826857086304/4782969), (79728367297319224858726778861520278606/4782969), (349411794076475502032284209435355526260/4782969)]
13
[0, 2, 4, 344, 8, 57132, 688, 9653632, 16, 1631462474, 114264, 275716983720, 1376, 46596170244988, 19307264, 7874752771571664, 32, 1330833218366359716, 3262924948, 224910813903914786296, 228528, 38009927549761627840864, 551433967440, 6423677755909710210314784, 2752, 1085601540748741025543476382, 93192340489976, 183466660386537233321693619440, 38614528, 31005865605324792430539070211100, 15749505543143328, 5239991287299889920761102865666160, 64]
[1, (2/1), (4/1), (120/1), (238/1), (11896/1), (30168/1), (1439184/1), (4157380/1), (189811274/1), (599285384/1), (26333803560/1), (88723902888/1), (49107536198680/13), (174014599298016/13), (7213347644514432/13), (26625378282277366/13), (1078499666025832432/13), (4118555809897238188/13), (163482775198224162600/13), (642659828981276272232/13), (25054722891994237364976/13), (100998502202797087904568/13), (3874362650104756230690576/13), (15967266110100116543774616/13), (603602510447504495417582514/13), (32981873521210096901958053048/169), (1230223091907221015523531389600/169), (5263220858135274477131152018576/169), (193918260012814317512428467441864/169), (843051531251554750427930738797968/169), (30710336237129574820405374448371728/169), (135481792315780160120520693827749956/169)]
14
[0, 2, 4, 398, 8, 76842, 796, 15059086, 16, 2951579306, 153684, 578509309974, 1592, 113387824750618, 30118172, 22224013651348518, 32, 4355906675618742306, 5903158612, 853757708421273485350, 307368, 167336510850569648301154, 1157018619948, 32797956126711642211745838, 3184, 6428399400835481873502559442, 226775649501236, 1259966282563754447215281083390, 60236344, 246953391382495871652459563712570, 44448027302697036, 48402864710969190843882074487652414, 64]
[1, (2/1), (4/1), (138/1), (274/1), (15910/1), (40392/1), (15622626/7), (45224476/7), (2388580466/7), (7560780980/7), (384155486874/7), (1298146709358/7), (63883151473270/7), (1589871232276560/49), (76150917694296330/49), (282102533838978994/49), (13199978916752542246/49), (50604616487570557204/49), (2319805329357741266214/49), (9157154376623375248550/49), (2885382773212778128378950/343), (11682271930773200713176432/343), (517319717015472950606545830/343), (2141799930639746180570959512/343), (93446351966709492620043432294/343), (394654778625977269826412659420/343), (16986782226515318977166145952558/343), (511245923937750634680376843722134/2401), (21732478167664507044633328583170790/2401), (94966769940751289768374782573582840/2401), (3990669153848631050678558933591570110/2401), (17698596203658532948686879177924398516/2401)]
15
[0, 2, 4, 456, 8, 101260, 912, 22781264, 16, 5125782618, 202520, 1153300781272, 1824, 259492675781276, 45562528, 58385852051087280, 32, 13136816711425781284, 10251565236, 2955783760070800781288, 405040, 665051346015930244128192, 2306601562544, 149636552853584289550781296, 3648, 33668224392056465148926287550, 518985351562552, 7575350488212704658523678129104, 91125056, 1704453859847858548164367675781308, 116771704102174560, 383502118465768173336982727050781312, 64]
[1, (2/1), (4/1), (472/3), (938/3), (313064/15), (2386504/45), (151209728/45), (438468628/45), (238769567458/405), (3786843695272/2025), (220335611053784/2025), (2239062417506024/6075), (126143302644447304/6075), (449684233739404016/6075), (369774174890311137728/91125), (1373860089823898950246/91125), (73558946314707859861232/91125), (2546044425707139966192508/820125), (133526162550137998667587832/820125), (2644261351956074515215785384/4100625), (408441242796375910761366669584/12301875), (1659560098368718343426033081848/12301875), (84044700417256080569197652438528/12301875), (1047768311200643964019372387039336/36905625), (1306791328925647158874157752613589818/922640625), (5540437459881500548817108743302380536/922640625), (7361317667821873690590736743132696619744/24911296875), (31777776362768149584931843040685933889664/24911296875), (1544177731895152755179938463490022632855256/24911296875), (101638604390204322392828632832673354742187984/373669453125), (4881698844674319275016799618600688602002361408/373669453125), (21743468643294774961891103968856101348218003956/373669453125)]
16
[0, 2, 4, 518, 8, 131082, 1036, 33554446, 16, 8589936146, 262164, 2199023255574, 2072, 562949953421338, 67108892, 144115188076251678, 32, 36893488147419103266, 17179872292, 9444732965739290427430, 524328, 2417851639229258450079274, 4398046511148, 618970019642690137449562158, 4144, 158456325028528675187088556082, 1125899906842676, 40564819207303340847920272380470, 134217784, 10384593717069655257060992658440250, 288230376152503356, 2658455991569831745807614120560689214, 64]
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17
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