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)]]
[5, 5, [1, 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250, 19531250, 97656250, 488281250, 2441406250, 12207031250, 61035156250, 305175781250, 1525878906250, 7629394531250, 38146972656250, 190734863281250, 953674316406250, 4768371582031250, 23841857910156250, 119209289550781250, 596046447753906250]]
[5, 6, [1, 12, 72, 388, 2064, (53868/5), (279736/5), (1441476/5), (7407264/5), (37905052/5), (968156712/25), (4932824004/25), (25102755504/25), (127519974396/25), (647219024568/25), (16403539991172/125), (83092235805504/125), (420496136844444/125), (2126824886907496/125), (10749092463251508/125), (271515077166847824/625), (1370822644105928828/625), (6918518650577048088/625), (34900178780979878772/625), (175999475132880624736/625), (22179596500657997790732/15625)]]
[5, 7, [1, 14, 98, 574, 3234, (88326/5), (475594/5), (2528438/5), (13342154/5), (69918534/5), (1823645138/25), (9471762062/25), (49042996114/25), (253179731742/25), (1304101077042/25), (33513609298246/125), (171960463437434/125), (880885292645078/125), (4506476637333594/125), (23024574857727814/125), (587564633702076674/625), (2995679531377744366/625), (15260083195014852642/625), (77668387769718423006/625), (395016799418092786466/625), (50189558159441392238974/15625)]]
[5, 8, [1, 16, 128, 816, 4864, (139024/5), (776064/5), (4256432/5), (23064064/5), (123766736/5), (3296540288/25), (17452929328/25), (91948495104/25), (482363960528/25), (2521597992832/25), (65704618352304/125), (341504398818304/125), (1770782486070992/125), (9163015516914304/125), (47325831834252656/125), (1220142572989869824/625), (6281958655913713104/625), (32299210226797923712/625), (165861550347107224304/625), (850764601262190567936/625), (108982625069781703061776/15625)]]
[6, 1, [1, 2, 2, (76/3), (146/3), (1700/3), (12212/9), (134632/9), (363382/9), (34654028/81), (101119420/81), (1039103992/81), (9620299796/243), (96438164776/243), (310631845816/243), (9150864961360/729), (30498567549782/729), (294262600837292/729), (9077850055174508/6561), (86260742395902808/6561), (302748416027244340/6561), (8515406375836001336/19683), (30498517789569080776/19683), (282549750248311094416/19683), (3089990698729416770428/59049), (28321342440570543682808/59049)]]
[6, 2, [1, 4, 8, (176/3), (608/3), (4288/3), (52352/9), (360704/9), (1593856/9), (97272832/81), (452036608/81), (3029504000/81), (43712798720/243), (290300149760/243), (1431794974720/243), (28317529997312/729), (142384007610368/729), (933000131117056/729), (42872594454020096/6561), (279497636842569728/6561), (1444971587678765056/6561), (28136609993004679168/19683), (146985684970825056256/19683), (950400864982384246784/19683), (15026783244602872692736/59049), (96836115263786158391296/59049)]]
[6, 3, [1, 6, 18, 108, 486, 2916, 14580, 87480, 459270, 2755620, 14880348, 89282088, 491051484, 2946308904, 16415149608, 98490897648, 554011299270, 3324067795620, 18836384175180, 113018305051080, 644204338791156, 3865226032746936, 22137203642096088, 132823221852576528, 763733525652315036, 4582401153913890216]]
[6, 4, [1, 8, 32, (544/3), (2816/3), (16256/3), (267776/9), (1560064/9), (8790016/9), (463456256/81), (2649137152/81), (15577538560/81), (269577420800/243), (1589441167360/243), (9226641080320/243), (163534318993408/729), (953684741586944/729), (5643412364066816/729), (297234855033307136/6561), (1761147060081393664/6561), (10334868127111708672/6561), (183900504873223847936/19683), (1081569054252140068864/19683), (6420927570355406503936/19683), (113496263941317273321472/59049), (674302167851734717693952/59049)]]
[6, 5, [1, 10, 50, (860/3), (4850/3), (28276/3), (490580/9), (2882120/9), (16848310/9), (894777340/81), (5261859292/81), (31138768280/81), (551414256980/243), (3269769175880/243), (19350617912440/243), (344751630687248/729), (2044103161905110/729), (12153347775892060/729), (649468449508131980/6561), (3865017054983936120/6561), (22975286291555444212/6561), (410489218360652392600/19683), (2442341389698467428360/19683), (14554489694679907960400/19683), (259984909698498160358140/59049), (1550128059942794919088600/59049)]]
[6, 6, [1, 12, 72, 432, 2592, 15552, 93312, 559872, 3359232, 20155392, 120932352, 725594112, 4353564672, 26121388032, 156728328192, 940369969152, 5642219814912, 33853318889472, 203119913336832, 1218719480020992, 7312316880125952, 43873901280755712, 263243407684534272, 1579460446107205632, 9476762676643233792, 56860576059859402752]]
[6, 7, [1, 14, 98, (1876/3), (11858/3), (73724/3), (1369844/9), (8426968/9), (51732982/9), (2847857684/81), (17396189692/81), (106022958664/81), (1936737427604/243), (11774584529176/243), (71536768009528/243), (1302404062357552/729), (7899828131463254/729), (47876222794915508/729), (2610284061425825708/6561), (15802432211281904104/6561), (95634906138094085812/6561), (1735393188350019944072/19683), (10493950086144704753224/19683), (63429241373853954098416/19683), (1149902230932570190241404/59049), (6946274672228368185753800/59049)]]
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