2018年5月27日日曜日

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)]]
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[8, 10, [0, 1, 8, 42, 192, 804, (16032/5), (61576/5), (1614848/35), (5925168/35), (3059328/5), (381919584/175), (14833769472/1925), (51900333888/1925), (180276361728/1925), (4354128819072/13475), (74707736788992/67375), (36437187049728/9625), (14740797728913408/1145375), (49911718246471168/1145375), (3200045314465316864/21762125), (53816222963078243328/108810625), (1263525787964402884608/761674375), (46492594833427858900992/8378418125), (509808714602480309895168/27529088125), (11882850479094194936983552/192703616875)]]
[8, 11, [0, 1, 8, 43, (608/3), (2635/3), (10904/3), (130571/9), (509056/9), (1945337/9), (65959240/81), (245355755/81), (9943914080/891), (108951967225/2673), (5137554878120/34749), (18515876957815/34749), (199164714110464/104247), (710606296663127/104247), (42928651911961784/1772199), (1367665004931889877/15949791), (4825795488729222880/15949791), (16974633249742477799/15949791), (10508872469700758888/2814669), (6876395197596171045421/526343103), (552196426994781972113536/12105891369), (5771380381970275795588075/36317674107)]]
[8, 12, [0, 1, 8, 44, (640/3), (2864/3), (61312/15), (760768/45), (21536768/315), (17088256/63), (3009501184/2835), (58176295936/14175), (222880104448/14175), (2540106862592/42525), (383473057792/1701), (251716282236928/297675), (14093131578343424/4465125), (52355496591818752/4465125), (3293842814716608512/75907125), (21860012600289132544/136632825), (7633365959810045968384/12980118375), (139868803741268147437568/64900591875), (10736991539290803099664384/1362912429375), (39151299102541683372326912/1362912429375), (28492348327518218456399872/272582485875), (1551908509329091693685768192/4088737288125)]]
[8, 13, [0, 1, 8, 45, 224, 1033, (22824/5), (97577/5), (571520/7), (11737107/35), (47581784/35), (136177013/25), (41646888288/1925), (32815556441/385), (8350946378312/25025), (227507801861763/175175), (4406368073763328/875875), (16991191157896259/875875), (1865103989184648/25025), (249898213773301447/875875), (18123009696687693536/16641625), (344870313403402674117/83208125), (9165623581068641023192/582456875), (76375535608216128176777/1281405125), (1443511190541887193580416/6407025625), (777940030197334866097711/915289375)]]
[8, 14, [0, 1, 8, 46, (704/3), (3340/3), (15200/3), (201080/9), (6077440/63), (25783280/63), (972023680/567), (4026312224/567), (16536558592/567), (202178230592/1701), (818952833536/1701), (23089277932160/11907), (277714977751040/35721), (1108607162026240/35721), (74965022000957440/607257), (2674496629064035840/5465313), (200862235702804398080/103840947), (791760997612155292672/103840947), (65387585277521735376896/2180659887), (256555037035254287091712/2180659887), (23107242755644033088094208/50155177401), (14259968870089456548352000/7919238537)]]
[8, 15, [0, 1, 8, 47, (736/3), (3587/3), (83896/15), (1142659/45), (5081216/45), (22221101/45), (863732968/405), (18454175047/2025), (78201190816/2025), (986840258747/6075), (53642418697336/78975), (223086661139857/78975), (13854237496019456/1184625), (57121617504477553/1184625), (234694323618869512/1184625), (8650195657579947703/10661625), (671192505343039879904/202570875), (13668913822737505236583/1012854375), (166639929499598264677352/3038563125), (675697460023041650074363/3038563125), (62896355357545734514281344/69886951875), (762243226481393842341247049/209660855625)]]
[8, 16, [0, 1, 8, 48, 256, 1280, 6144, 28672, 131072, 589824, 2621440, 11534336, 50331648, 218103808, 939524096, 4026531840, 17179869184, 73014444032, 309237645312, 1305670057984, 5497558138880, 23089744183296, 96757023244288, 404620279021568, 1688849860263936, 7036874417766400]]

180527(3)

PARI


Expansion of ((1 + k*x)/(1 - k*x))^(m/k)(3)

具体例で確認してみよう。

a(0) = 0, a(1) = 1
n > 1 に対し、
a(n) = 3*a(n-1)/(n-1) + 2*a(n-2)
で定められる数列{a(n)} の母関数は
x/3 * d/dx ((1 + √2*x)/(1 - √2*x))^(3/(2*√2))
となるはずである。

まず、定義通りa(n) を求めることにする。
a(0) ~ a(25) の値は以下の通り。
[0.0, 1.0, 3.0, 6.5, 12.5, 22.375, 38.425, 63.9625, 104.2625, 167.0234375, 264.19947916666666, 413.30671875, 641.1189725378788, 986.8931806344697, 1509.9825252221738, 2297.3540452451193, 3479.435859493371, 5247.102314145246, 7884.830950894727, 11808.34312010628, 17634.13713127992, 26261.80680990455, 39019.960949689055, 57844.51738567579, 85584.85894968365, 126387.14214006203]

次に母関数からa(n) を求めてみる。

(17:03) gp > N=26; x='x+O('x^N); Vec(x/(1-2^(1/2)*x)^2 * ((1+2^(1/2)*x)/(1-2^(1/2)*x))^(3/(2^(3/2))-1))
%1 = [1, 3.0000000000000000000000000000000000000, 6.5000000000000000000000000000000000000, 12.500000000000000000000000000000000000, 22.375000000000000000000000000000000000, 38.425000000000000000000000000000000000, 63.962500000000000000000000000000000001, 104.26250000000000000000000000000000000, 167.02343750000000000000000000000000000, 264.19947916666666666666666666666666667, 413.30671875000000000000000000000000000, 641.11897253787878787878787878787878788, 986.89318063446969696969696969696969697, 1509.9825252221736596736596736596736597, 2297.3540452451194638694638694638694638, 3479.4358594933712121212121212121212120, 5247.1023141452460300116550116550116547, 7884.8309508947270177738927738927738924, 11808.343120106279896318958818958818958, 17634.137131279919282334989571831677093, 26261.806809904547684988166073692389480, 39019.960949689059662525431439905124111, 57844.517385675785323957072798280932247, 85584.858949683656541219176723064282863, 126387.14214006202771556654268694489985]
(17:04) gp >

確かに一致する。

180527(2)

Expansion of ((1 + k*x)/(1 - k*x))^(m/k)(2)

a(0) = 0, a(1) = 1
n > 1 に対し、
a(n) = 2*m*a(n-1)/(n-1) + k^2*a(n-2)
で定められる数列{a(n)} の母関数は
x/(2*m) * d/dx ((1 + k*x)/(1 - k*x))^(m/k)
であることがわかる。

すなわち、
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)} の母関数は
x/s * d/dx ((1 + √t*x)/(1 - √t*x))^(s/(2*√t))
であることがわかる。

180527

Ruby


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

((1 + k*x)/(1 - k*x))^(m/k) = a(0) + a(1)*x + a(2)*x^2 + ...
とする。このとき、n > 1 に対し、
n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2)
が成り立つ。
このことを利用し、a(n) 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..8).each{|i|
  (1..8).each{|j|
    p [i, j, A(i, j, n)]
  }
}

出力結果
[1, 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]]
[1, 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]]
[1, 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]]
[1, 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]]
[1, 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]]
[1, 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]]
[1, 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]]
[1, 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]]
[2, 1, [1, 2, 2, 4, 6, 12, 20, 40, 70, 140, 252, 504, 924, 1848, 3432, 6864, 12870, 25740, 48620, 97240, 184756, 369512, 705432, 1410864, 2704156, 5408312]]
[2, 2, [1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864]]
[2, 3, [1, 6, 18, 44, 102, 228, 500, 1080, 2310, 4900, 10332, 21672, 45276, 94248, 195624, 404976, 836550, 1724580, 3549260, 7293000, 14965236, 30669496, 62783448, 128388624, 262303132, 535422888]]
[2, 4, [1, 8, 32, 96, 256, 640, 1536, 3584, 8192, 18432, 40960, 90112, 196608, 425984, 917504, 1966080, 4194304, 8912896, 18874368, 39845888, 83886080, 176160768, 369098752, 771751936, 1610612736, 3355443200]]
[2, 5, [1, 10, 50, 180, 550, 1532, 4020, 10120, 24710, 58940, 138012, 318360, 725340, 1635480, 3655080, 8106384, 17859270, 39116220, 85230860, 184853240, 399257716, 859115400, 1842353240, 3938662000, 8396404380, 17852837912]]
[2, 6, [1, 12, 72, 304, 1056, 3264, 9344, 25344, 66048, 166912, 411648, 995328, 2367488, 5554176, 12877824, 29556736, 67239936, 151781376, 340262912, 758120448, 1679818752, 3703570432, 8128561152, 17767071744, 38688260096, 83953188864]]
[2, 7, [1, 14, 98, 476, 1862, 6356, 19796, 57752, 160454, 429268, 1114428, 2823240, 7008540, 17103240, 41132520, 97681584, 229435206, 533705172, 1230873644, 2817062248, 6403088692, 14463808216, 32488200472, 72599769424, 161473267228, 357592181128]]
[2, 8, [1, 16, 128, 704, 3072, 11520, 38912, 121856, 360448, 1019904, 2785280, 7389184, 19136512, 48562176, 121110528, 297533440, 721420288, 1729101824, 4102029312, 9642704896, 22481469440, 52026146816, 119587995648, 273200185344, 620622774272, 1402575257600]]
[3, 1, [1, 2, 2, (22/3), (38/3), (134/3), (818/9), (2818/9), (6226/9), (189986/81), (441442/81), (1479250/81), (10672070/243), (35437030/243), (87389830/243), (864182434/729), (2172607538/729), (7118226098/729), (163545968834/6561), (533099435506/6561), (1378032291106/6561), (13416581150534/19683), (35044118159014/19683), (113296611902998/19683), (895666077411346/59049), (2885941125863378/59049)]]
[3, 2, [1, 4, 8, (68/3), (176/3), (508/3), (4184/9), (12188/9), (34336/9), (905188/81), (2587048/81), (7606220/81), (65814800/243), (194023580/243), (563149480/243), (4990671356/729), (14552074304/729), (43055819428/729), (1133860988744/6561), (3359121437228/6561), (9856098296272/6561), (87690594135868/19683), (257866157115016/19683), (765434213919092/19683), (6764904495556192/59049), (20095770593039236/59049)]]
[3, 3, [1, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886]]
[3, 4, [1, 8, 32, (328/3), (1088/3), (3512/3), (33632/9), (106168/9), (333184/9), (9354056/81), (29073568/81), (90024280/81), (834203840/243), (2570064760/243), (7903895200/243), (72785747704/729), (223122397952/729), (683003242568/729), (18796825622816/6561), (57414214210456/6561), (175219973228992/6561), (1602798630831416/19683), (4883689754104864/19683), (14869498228694632/19683), (135740819642790016/59049), (412795398286467464/59049)]]
[3, 5, [1, 10, 50, (590/3), (2150/3), (7486/3), (76130/9), (253130/9), (830290/9), (24250090/81), (78052882/81), (249526010/81), (2380004870/243), (7531482590/243), (23739667990/243), (223716028586/729), (700672174130/729), (2188728564730/729), (61392039361010/6561), (190936822486490/6561), (592743930067426/6561), (5511090849409150/19683), (17054210487750070/19683), (52701664148514350/19683), (487968789757457170/59049), (1504296853352079322/59049)]]
[3, 6, [1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300]]
[3, 7, [1, 14, 98, (1498/3), (6566/3), (26474/3), (303506/9), (1117582/9), (4004434/9), (126469742/81), (436544962/81), (1486879870/81), (15026341190/243), (50151699850/243), (166069189030/243), (1638543505774/729), (5357110158386/729), (17423700911582/729), (507677837784866/6561), (1636838204433502/6561), (5257977229160866/6561), (50501573452340138/19683), (161196806003619526/19683), (513110985501867802/19683), (4887565173217851922/59049), (15482713376868393278/59049)]]
[3, 8, [1, 16, 128, (2192/3), (10496/3), (45424/3), (552320/9), (2138480/9), (8005120/9), (262806160/81), (939221632/81), (3301349552/81), (34337884928/243), (117685151984/243), (399389571712/243), (4031879185904/729), (13467457817600/729), (44693118539920/729), (1327201911186560/6561), (4356718674129200/6561), (14235710419914496/6561), (138967179999246448/19683), (450489023033716864/19683), (1455331364712915152/19683), (14060266049510322688/59049), (45149001371155418896/59049)]]
[4, 1, [1, 2, 2, 12, 22, 124, 276, 1496, 3686, 19436, 51068, 263720, 724860, 3681880, 10466920, 52450992, 153093254, 758495564, 2261603564, 11096526344, 33676743956, 163842737928, 504738342808, 2437418983888, 7605947276508, 36487283224952]]
[4, 2, [1, 4, 8, 32, 96, 384, 1280, 5120, 17920, 71680, 258048, 1032192, 3784704, 15138816, 56229888, 224919552, 843448320, 3373793280, 12745441280, 50981765120, 193730707456, 774922829824, 2958796259328, 11835185037312, 45368209309696, 181472837238784]]
[4, 3, [1, 6, 18, 68, 246, 948, 3572, 13896, 53286, 208452, 807132, 3169080, 12346300, 48602760, 190150440, 750018448, 2943363078, 11627329764, 45736940364, 180897649368, 712881236052, 2822389182104, 11138924119512, 44137230865392, 174405194802524, 691557285091176]]
[4, 4, [1, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832, 140737488355328, 562949953421312, 2251799813685248]]
[4, 5, [1, 10, 50, 220, 950, 4012, 16820, 69880, 289190, 1190940, 4892572, 20038280, 81932860, 334312760, 1362445480, 5544100592, 22539299590, 91528066940, 371407853580, 1505774354920, 6101160269012, 24703190789800, 99972872453720, 404347862170000, 1634747071892060, 6605899359899224]]
[4, 6, [1, 12, 72, 352, 1632, 7296, 32000, 138240, 591360, 2508800, 10579968, 44384256, 185450496, 772079616, 3205103616, 13270253568, 54824140800, 226044149760, 930417213440, 3823632384000, 15692187303936, 64318594875392, 263332867080192, 1077001838395392, 4400716303040512, 17965810886639616]]
[4, 7, [1, 14, 98, 532, 2646, 12516, 57428, 257896, 1140454, 4983412, 21574588, 92695960, 395806460, 1681213800, 7109416680, 29948286928, 125736584582, 526347120404, 2197634741036, 9154384374968, 38054009333396, 157889951459896, 653988286687512, 2704646074093488, 11169538414638044, 46067331722853448]]
[4, 8, [1, 16, 128, 768, 4096, 20480, 98304, 458752, 2097152, 9437184, 41943040, 184549376, 805306368, 3489660928, 15032385536, 64424509440, 274877906944, 1168231104512, 4947802324992, 20890720927744, 87960930222080, 369435906932736, 1548112371908608, 6473924464345088, 27021597764222976, 112589990684262400]]
[5, 1, [1, 2, 2, 18, 34, (1418/5), (3306/5), (26266/5), (68554/5), (525962/5), (7381362/25), (55133634/25), (162967314/25), (1191360306/25), (3662351058/25), (131505600522/125), (417007847034/125), (2949918581754/125), (9594609776506/125), (66994979831498/125), (1112891089772674/625), (7682802822826338/625), (25991415933272258/625), (177628448428277042/625), (610438985839845666/625), (103357235817938990482/15625)]]
[5, 2, [1, 4, 8, 44, 144, (3876/5), (14584/5), (77548/5), (312224/5), (1646644/5), (34515688/25), (180957932/25), (779396144/25), (4067770452/25), (17863566072/25), (464493220396/125), (2069950844224/125), (10733221236788/125), (48384067924264/125), (250271331438524/125), (5693478972918224/625), (29388968002769876/625), (134740698112281432/625), (694268304082752636/625), (3203519049086908256/625), (412018351043930398724/15625)]]
[5, 3, [1, 6, 18, 86, 354, (8574/5), (38074/5), (185742/5), (853194/5), (4180446/5), (97860738/25), (480924198/25), (2279227474/25), (11225347638/25), (53651452002/25), (1323382231454/125), (6364395899514/125), (31438512481902/125), (151910524149834/125), (751201628931966/125), (18216736410254274/625), (90162108913092614/625), (438606402663895122/625), (2172467199798130134/625), (10594513527663795746/625), (1312735721049907601526/15625)]]
[5, 4, [1, 8, 32, 152, 704, (17032/5), (81376/5), (397144/5), (1922944/5), (9431528/5), (230020512/25), (1131875736/25), (5546677824/25), (27356865384/25), (134489876448/25), (3322300087128/125), (16370980280064/125), (80990022053736/125), (399795127136416/125), (1979953704733112/125), (48936859212313024/625), (242565948687602152/625), (1200407145257151328/625), (5954364357523840888/625), (29494118531317664896/625), (3659712453826749829768/15625)]]
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2018年5月1日火曜日

180501

分割の逆数和が1(2)

「77より大きな整数は、適当な分割が存在して、分割した数の逆数(単位分数)の和を1にすることができる。」
ということは、A028229 によればGraham によって示されたようだ。