180527(4)

Ruby


Expansion of ((1 + k*x)/(1 - k*x))^(m/k)(4)

s, t を正の数とし、
a(0) = 0, a(1) = 1
n > 1 に対し、
a(n) = s*a(n-1)/(n-1) + t*a(n-2)
で定められる数列{a(n)} を出力してみた。

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

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

出力結果
[1, 1, [0, 1, 1, (3/2), (3/2), (15/8), (15/8), (35/16), (35/16), (315/128), (315/128), (693/256), (693/256), (3003/1024), (3003/1024), (6435/2048), (6435/2048), (109395/32768), (109395/32768), (230945/65536), (230945/65536), (969969/262144), (969969/262144), (2028117/524288), (2028117/524288), (16900975/4194304)]]
[1, 2, [0, 1, 1, (5/2), (17/6), (137/24), (817/120), (9037/720), (2219/144), (155687/5760), (1753367/51840), (29777027/518400), (415517767/5702400), (1655330579/13685760), (137918196199/889574400), (450088549997/1779148800), (8725180321937/26687232000), (224767684320497/426995712000), (11697095951657/17079828480), (8384064034329317/7685922816000), (208404404807664017/146032533504000), (598208460990722267/265513697280000), (25948570444190817031/8761952010240000), (178909451160543909743/38552588845056000), (1180630458492340277569/192762944225280000), (2322047301948572558731/243490034810880000)]]
[1, 3, [0, 1, 1, (7/2), (25/6), (277/24), (1777/120), (26707/720), (250609/5040), (947477/8064), (8409847/51840), (1337962879/3628800), (1887700859/3628800), (50054364503/43545600), (14361513331/8709120), (2174090875781/609638400), (47412857868431/9144576000), (230394041204393/20901888000), (40301650309070447/2487324672000), (304162751817699973/8954368819200), (42870306976194778487/850665037824000), (1776597992337084624587/17013300756480000), (55793184782342505518207/357279315886080000), (228923454741958345017799/714558631772160000), (579348407358803369129/1200938876928000), (16827201043799488845913283/17149407162531840000)]]
[1, 4, [0, 1, 1, (9/2), (11/2), (155/8), (207/8), (1309/16), (1843/16), (43731/128), (63835/128), (362615/256), (543645/256), (5983055/1024), (9158555/1024), (49172805/2048), (76546627/2048), (3223606147/32768), (5088608019/32768), (26354250067/65536), (42095929945/65536), (430087187061/262144), (694015221361/262144), (3503789789339/524288), (5704460457381/524288), (114022760077975/4194304)]]
[1, 5, [0, 1, 1, (11/2), (41/6), (701/24), (4801/120), (109951/720), (1118161/5040), (31904441/40320), (434442401/362880), (14791440851/3628800), (23066796491/3628800), (910553247551/43545600), (18902654510531/566092800), (847506109781941/7925299200), (20695293345839491/118879488000), (1037702625084168691/1902071808000), (29183301575425876451/32335220736000), (1616868317954203973681/582033973248000), (2711595474312234352889/582033973248000), (164398427269732631720989/11640679464960000), (5858748923325424772787889/244454268764160000), (35056283265127982186206589/488908537528320000), (1382568535629975679927421059/11244896363151360000), (98137910347383206513857606699/269877512715632640000)]]
[1, 6, [0, 1, 1, (13/2), (49/6), (985/24), (1373/24), (36833/144), (382829/1008), (12758717/8064), (178140845/72576), (201938515/20736), (24928161145/1596672), (1144475288305/19160064), (24477234120025/249080832), (1274244248949085/3487131648), (494085526772009/804722688), (373405455545372597/167382319104), (1550846284505759363/406499917824), (696428340372844403633/51218989645824), (1209093914263872204935/51218989645824), (16956098951801040128179/204875958583296), (626339431740792631415419/4302395130249216), (47628645726133275866727607/94652692865482752), (84747615673529552384519117/94652692865482752), (991896085748103039599042075/324523518395940864)]]
[1, 7, [0, 1, 1, (15/2), (19/2), (439/8), (3099/40), (31763/80), (67093/112), (12786561/4480), (20206769/4480), (915266039/44800), (16474478169/492800), (57478686547/394240), (6284103486251/25625600), (372423336790641/358758400), (3203351820526537/1793792000), (211760420423285497/28700672000), (1262188457341162389/97582284800), (51100195870375927391/975822848000), (91042676007059047619/975822848000), (50664825859158663513/136478720000), (91636845852871096294073/136615198720000), (316072896453824628459083/120221374873600), (332479530422215038785083641/69127290552320000), (10288373775953891382644167147/553018324418560000)]]
[1, 8, [0, 1, 1, (17/2), (65/6), (1697/24), (12097/120), (419377/720), (640567/720), (5496139/1152), (396447287/51840), (20182547687/518400), (369056160247/5702400), (21681826517719/68428800), (19290556580239/35582976), (4578714760614977/1779148800), (120322054242048977/26687232000), (685496491894061909/32845824000), (15925426142665963457/426995712000), (259834971793669971421/1537184563200), (4079194616845675079587/13275684864000), (3994362712049085991474657/2920650670080000), (22108770821523605276144311/8761952010240000), (2131132282783441008774763207/192762944225280000), (796760317985313001448755949/38552588845056000), (413161199884347238691998315489/4626310661406720000)]]
[1, 9, [0, 1, 1, (19/2), (73/6), (2125/24), (3053/24), (117803/144), (181691/144), (8663507/1152), (126399275/10368), (1428767989/20736), (2405074949/20736), (156712017761/248832), (271802865089/248832), (2859645300425/497664), (15249283774891/1492992), (1250616053558491/23887872), (2269462513793627/23887872), (204869263190269169/429981696), (378435520034055425/429981696), (7450980578856501169/1719926784), (41935461960657486067/5159780352), (406165265981947198223/10319560704), (772497674682354192607/10319560704), (88504195126782949008775/247669456896)]]
[1, 10, [0, 1, 1, (21/2), (27/2), (867/8), (6267/40), (88789/80), (966169/560), (50688009/4480), (11846503/640), (5151726421/44800), (96369799521/492800), (2298882891747/1971200), (4031629126359/1971200), (325875233970939/27596800), (2930765466441613/137984000), (37661564663313259/315392000), (8235313021364380173/37531648000), (90549558678837154217/75063296000), (3219968506797301619957/1426202624000), (347308291486378487644557/28524052480000), (271983843530098927283607/11745198080000), (539478494229034552736439099/4392704081920000), (3419361244955449182524616177/14433170554880000), (1000618938952986291794272046573/808257551073280000)]]
[1, 11, [0, 1, 1, (23/2), (89/6), (3125/24), (4541/24), (210791/144), (329819/144), (18879427/1152), (280096075/10368), (3794145761/20736), (71577395911/228096), (5580677040883/2737152), (128407488424159/35582976), (1614417560610275/71165952), (8797777748116549/213497856), (861210249750341749/3415965696), (27184161272115056357/58071416832), (2926017861931765383491/1045285502976), (5536464871980453020975/1045285502976), (129852078899393767477799/4181142011904), (749363660087047479836957/12543426035712), (95021972941046922668719031/275955372785664), (4265980104985553195441221693/6346973574070272), (581239399803022467639903177925/152327365777686528)]]
[1, 12, [0, 1, 1, (25/2), (97/6), (3697/24), (26977/120), (1357897/720), (2990861/1008), (927461089/40320), (13847980609/362880), (1015505956729/3628800), (1754076396419/3628800), (29597386833079/8709120), (263970611404751/43545600), (25125775551191111/609638400), (690331716291163631/9144576000), (73052565303721563311/146313216000), (465259057455615930979/497464934400), (270575315082543660132887/44771844096000), (9817691174071782563821967/850665037824000), (1243641127950470872769786687/17013300756480000), (50724804645272254994432500367/357279315886080000), (126281294872776050429638725329/142911726354432000), (1244847766893659348220649296923/714558631772160000), (10769994846099480703935318457099/1008788656619520000)]]
[1, 13, [0, 1, 1, (27/2), (35/2), (1439/8), (10539/40), (190583/80), (2108681/560), (28170621/896), (234953169/4480), (2649408117/6400), (354528888489/492800), (10726406396631/1971200), (50077587003039/5125120), (25629065469444141/358758400), (236396042686975497/1793792000), (26890624130908882137/28700672000), (7250311185479447391/4100096000), (140959071665134101151/11480268800), (25776371437713599583833/1090625536000), (318860494687865990728503/1982955520000), (48082151163827593208580571/152687575040000), (7070027965180012441031673637/3359126650880000), (562355510123196383594912469/134365066035200), (105709886328059228151988332689/3839001886720000)]]
[1, 14, [0, 1, 1, (29/2), (113/6), (4985/24), (7325/24), (426065/144), (4733165/1008), (338768125/8064), (5109798445/72576), (86391527189/145152), (150928131659/145152), (14664704699411/1741824), (26483980326359/1741824), (2900766101410915/24385536), (16152733652181275/73156608), (1965467553800316155/1170505728), (63475077301306611355/19898597376), (8483538077781861019375/358174752768), (312402208196437916187115/6805320302592), (9088964956399187707852423/27221281210368), (376473961795410177143899663/571646905417728), (5378536299980486933775761057/1143293810835456), (247827767696224641014447144029/26295757649215488), (41813156293945427665233528592525/631098183581171712)]]
[1, 15, [0, 1, 1, (31/2), (121/6), (5701/24), (42001/120), (2607451/720), (4152583/720), (317046703/5760), (4801836343/51840), (432814885393/518400), (759622259213/518400), (78666301629953/6220800), (1856182388188373/80870400), (30945026548280009/161740800), (866227101233047859/2426112000), (112268322675041080259/38817792000), (214498523276816255587/38817792000), (30526945645537907925517/698720256000), (1130904370055605299086827/13275684864000), (175134494549621680474533727/265513697280000), (1042833146557133580674506261/796541091840000), (15856907522789327022769354181/1593082183680000), (735411778647211497688178674271/36640890224640000), (132030606067342839246218431292951/879381365391360000)]]
[1, 16, [0, 1, 1, (33/2), (43/2), (2155/8), (3183/8), (70021/16), (111859/16), (9074547/128), (15326235/128), (293450751/256), (517116861/256), (18953220351/1024), (34553419131/1024), (611439253965/2048), (1146472029123/2048), (157674921044163/32768), (302771834811027/32768), (5079238788392219/65536), (9956027071236665/65536), (327062487871349349/262144), (652760136743496529/262144), (10525341442496224307/524288), (21345947916769985637/524288), (1354359020611773373175/4194304)]]
[2, 1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]]
[2, 2, [0, 1, 2, 4, (20/3), (34/3), (268/15), (1288/45), (1976/45), (614/9), (41708/405), (318008/2025), (5223896/22275), (23600476/66825), (90932968/173745), (678564496/868725), (14997074192/13030875), (22231569154/13030875), (32609627108/13030875), (86555574376/23455575), (12018048214696/2228279625), (6787447674196/857030625), (385751966373608/33424194375), (6209382070833776/367666138125), (361059581089936/14706645525), (39512914806814756/1102998414375)]]
[2, 3, [0, 1, 2, 5, (28/3), (59/3), (538/15), (3193/45), (8056/63), (77123/315), (1241806/2835), (1664773/2025), (20745892/14175), (23050729/8505), (204444358/42525), (2624770903/297675), (69649514576/4465125), (126820879957/4465125), (108736200094/2168775), (62016550903553/683164125), (2076391574494876/12980118375), (18712912794760043/64900591875), (691489171555406026/1362912429375), (49671046321178603/54516497175), (2182448050147041128/1362912429375), (1674513919257420851/584105326875)]]
[2, 4, [0, 1, 2, 6, 12, 30, 60, 140, 280, 630, 1260, 2772, 5544, 12012, 24024, 51480, 102960, 218790, 437580, 923780, 1847560, 3879876, 7759752, 16224936, 32449872, 67603900]]
[2, 5, [0, 1, 2, 7, (44/3), (127/3), (1354/15), (10879/45), (163928/315), (421747/315), (1174322/405), (103113329/14175), (224254228/14175), (1658827049/42525), (47047228558/552825), (801813535853/3869775), (26303422064656/58046625), (1294366182593/1184625), (2362638761389874/986792625), (50881955115888479/8881133625), (111674739537901012/8881133625), (1327886247666162481/44405668125), (61285010752730356262/932519030625), (144999420618831638347/932519030625), (1048253582543093463832/3063991100625), (51859243882947328791421/64343813113125)]]
[2, 6, [0, 1, 2, 8, (52/3), (170/3), (380/3), (3440/9), (54760/63), (158170/63), (3273380/567), (9195856/567), (234434792/6237), (1937996884/18711), (58733735096/243243), (1116880033760/1702701), (7847202635600/5108103), (1621903149010/392931), (842584150705460/86837751), (20198376330990800/781539759), (47625689015041240/781539759), (11450256989768084/71049069), (367808380568829064/965431467), (180823360535674302752/180535684329), (428404773479589192656/180535684329), (3361921683012034747700/541607052987)]]
[2, 7, [0, 1, 2, 9, 20, 73, (846/5), (2837/5), (47128/35), (30159/7), (363406/35), (5641231/175), (151193772/1925), (459573749/1925), (587111230/1001), (307426258863/175175), (3801007122992/875875), (11235044950579/875875), (474789937537206/14889875), (277944957324607/2977975), (3469816382404972/14889875), (50375275723008711/74449375), (883688530837890614/521145625), (28035962145539585843/5732601875), (324216862480996689816/26369968625), (4648880264798955274813/131849843125)]]
[2, 8, [0, 1, 2, 10, (68/3), (274/3), (3268/15), (36148/45), (17752/9), (311374/45), (7013468/405), (119108108/2025), (3324142136/22275), (6621322316/13365), (1103345569592/868725), (3600708399976/868725), (139602885150992/13030875), (449535368640994/13030875), (46788383806628/521235), (33536256137317268/117277875), (1667235238461312136/2228279625), (2392833843962889068/1012854375), (207588563553526536248/33424194375), (1431275609284351277944/73533227625), (18890087335877444441104/367666138125), (9288189207794290234924/58052548125)]]
[2, 9, [0, 1, 2, 11, (76/3), (335/3), (818/3), (9863/9), (24904/9), (94993/9), (2207210/81), (8135875/81), (21344140/81), (230340695/243), (611728810/243), (2160456085/243), (17380860304/729), (60504921833/729), (163545968834/729), (5064444637307/6561), (13780322911060/6561), (46958034026869/6561), (385485299749154/19683), (1302911036884477/19683), (3582664309645384/19683), (36074264073292225/59049)]]
[2, 10, [0, 1, 2, 12, 28, 134, (1668/5), (7256/5), (131272/35), (540738/35), (1432884/35), (4067112/25), (845025768/1925), (3272513868/1925), (8953721352/1925), (238029692112/13475), (3292488934608/67375), (12313045722426/67375), (83478458618316/163625), (2158145462848888/1145375), (10485694626187096/1978375), (2107909510150472628/108810625), (41775197317587301352/761674375), (1664865520133451224912/8378418125), (109020980253762774870384/192703616875), (56000593045501049471756/27529088125)]]
[2, 11, [0, 1, 2, 13, (92/3), (475/3), (1202/3), (16877/9), (44488/9), (196769/9), (4797850/81), (20439701/81), (621419252/891), (7730321089/2673), (282049501286/34749), (1145728701625/34749), (751225001776/8019), (39029787781511/104247), (1904287554880478/1772199), (67591420391163491/15949791), (195639354290131900/15949791), (763069559731811591/15949791), (6674118565783441726/47849373), (283668368748431049259/526343103), (19141408706072180421976/12105891369), (220089644056577211493075/36317674107)]]
[2, 12, [0, 1, 2, 14, (100/3), (554/3), (7108/15), (106828/45), (2004872/315), (1894954/63), (33639388/405), (5351851516/14175), (15101606872/14175), (200217458012/42525), (114892106648/8505), (17392727006248/297675), (758605725894896/4465125), (460788082408786/637875), (161206601236281788/75907125), (1216651007270799892/136632825), (342962455809558227896/12980118375), (7106391969348338498348/64900591875), (446345478258740044145656/1362912429375), (1831387637935666760142392/1362912429375), (9269574517740853906064/2290609125), (67308804175197955383653132/4088737288125)]]
[2, 13, [0, 1, 2, 15, 36, 213, (2766/5), (14767/5), (56248/7), (1414107/35), (3970366/35), (95887321/175), (3030586332/1925), (568679385/77), (540603059358/25025), (17359295870733/175175), (257547255921712/875875), (1160547638587859/875875), (696929868657342/175175), (15474302562007357/875875), (891656992915832084/16641625), (19556592160537001937/83208125), (418741659883728266178/582456875), (3998978082516253441023/1281405125), (61618743486206295733464/6407025625), (265068470654073664977617/6407025625)]]
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