240414

PARI


Apéry numberの仲間たち

l>=7 のとき、closed formはまだ見つかっていないようだ。

h(n) = sum(k=1, n, 1/k);
a(n, l) = sum(k=0, n, binomial(n, k)^l*(1-l*k*(h(k)-h(n-k))));
for(l=1, 12, for(n=0, 10, print1(a(n, l), ", ")); print)

出力結果

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

1, -2, 6, -20, 70, -252, 924, -3432, 12870, -48620, 184756, 

1, -3, 19, -147, 1251, -11253, 104959, -1004307, 9793891, -96918753, 970336269, 

1, -4, 48, -760, 13840, -273504, 5703096, -123519792, 2751843600, -62659854400, 1451780950048, 

1, -5, 109, -3317, 121501, -4954505, 216867925, -9981053045, 476860000285, -23451310381505, 1180189308268609, 

1, -6, 234, -13164, 936810, -76041756, 6739851636, -635746866264, 62819253878634, -6434646802174140, 678284981981948484, 

1, -7, 487, -49255, 6669751, -1053222757, 184786011019, -34916924925799, 6974335808409655, -1454249641139993257, 313810538382057898237, 

1, -8, 996, -177200, 45086980, -13631185008, 4662729892464, -1737911032294368, 690950882254608900, -288779498502632656640, 125605455885282444138496, 

1, -9, 2017, -620073, 294290001, -168314616009, 111074551176961, -80766140154871593, 63220243361206024657, -52382852763175965870009, 45422849668460020450930017, 

1, -10, 4062, -2125828, 1874792350, -2009033203260, 2538914209085100, -3571944365364382920, 5460901555410763324830, -8898003584839672562442700, 15258440639018489319278045812,

240330

PARI


Continued fraction for Pi^n.

Pi^4の連分数展開が興味深い。

default(realprecision, 70);
for(n=1, 100, print([n, contfrac(Pi^n)]))

出力結果

[1, [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4]]

[2, [9, 1, 6, 1, 2, 47, 1, 8, 1, 1, 2, 2, 1, 1, 8, 3, 1, 10, 5, 1, 3, 1, 2, 1, 1, 3, 15, 1, 1, 2, 2, 1, 3, 2, 7, 1, 9, 18, 30, 2, 145, 1, 1, 17, 9, 1, 1, 1, 1, 7, 12, 1, 2, 1, 12, 1, 1, 4, 1, 5, 1, 1, 2, 3, 4, 1, 3, 2, 9, 1, 20, 11, 14]]

[3, [31, 159, 3, 7, 1, 13, 2, 1, 3, 1, 12, 2, 2, 4, 34, 2, 43, 3, 1, 3, 2, 1, 1, 5, 1, 1, 4, 1, 5, 4, 2, 4, 11, 3, 3, 1, 1, 2, 1, 7, 2, 1, 1, 3, 1, 12, 3, 1, 9, 2, 1, 8, 23, 1, 45, 1, 1, 2, 1, 23, 3, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 16]]

[4, [97, 2, 2, 3, 1, 16539, 1, 6, 7, 6, 8, 6, 3, 9, 1, 1, 1, 18, 1, 4, 1, 13, 1, 2, 1, 127, 1, 1, 1, 4, 1, 6, 1, 1, 1, 10, 10, 1, 1, 2, 1, 2, 1, 5, 1, 1, 10, 1, 3, 2, 1, 1, 4, 9, 1, 7, 70, 1, 13, 1, 2, 6, 1, 2, 24, 5, 2]]

[5, [306, 50, 1, 4, 60, 1, 4, 23, 1, 2, 1, 1, 1, 17, 1, 5, 1, 1, 1, 2, 4, 10, 4, 1, 3, 2, 1, 36, 7, 1, 27, 3, 1, 6, 1, 2, 1, 1, 3, 1, 1, 1, 53, 1, 1, 1, 15, 1, 3, 573, 1, 5, 2, 1, 4, 1, 3, 2, 1, 15, 2, 1, 1, 1, 4, 3, 1, 66]]

[6, [961, 2, 1, 1, 3, 9, 1, 5, 1, 3, 6, 1, 3, 1, 77, 11, 3, 1, 11, 14, 4, 4, 27, 5, 178, 5, 1, 2, 2, 1099, 2, 3, 71, 4, 1, 3, 1, 1, 1, 2, 16, 154, 1, 3, 1, 5, 4, 5, 1, 1, 17, 1, 1, 1, 1, 1, 2, 4]]

[7, [3020, 3, 2, 2, 3, 2, 10, 4, 1, 1, 2, 1, 3, 13, 58, 2, 50, 1, 2, 21, 1, 1, 2, 1, 14, 1, 182, 2, 1, 1, 2, 69, 6, 2, 4, 3, 2, 5, 11, 2, 2, 5, 4, 1, 5, 41, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 8, 1, 5, 1, 2, 2, 3]]

[8, [9488, 1, 1, 7, 1, 1, 3, 1, 1, 1, 4, 10, 13, 1, 1, 2, 3, 6, 1, 1, 1, 135, 1, 2, 2, 3, 1, 2, 1, 7, 3, 1, 1, 4, 2, 7, 1, 1, 1, 27, 4, 1, 3, 13, 1, 3, 2, 17, 1, 1, 3, 3, 2, 1, 1, 1, 1, 1, 6, 5, 5, 5, 1, 2, 1, 1, 1, 2, 2, 2, 1, 6, 11, 2, 24, 3, 25, 3]]

[9, [29809, 10, 14, 1, 9, 3, 1, 5, 8, 16, 3, 15, 7, 39, 3, 3, 1, 1, 2, 1, 3, 4, 1, 1, 7, 2, 1, 2, 1, 3, 10, 5, 72, 1, 1, 3, 2, 1, 2, 2, 6, 3, 7, 2, 1, 3, 6, 1, 6, 2, 3, 20, 1, 2, 8, 9, 1, 14, 3, 1, 2, 4, 14]]

[10, [93648, 21, 15, 1, 4, 2, 1, 3, 2, 1, 4, 2, 74, 26, 5, 3, 1, 5, 17, 1, 2, 1, 1, 2, 1, 14, 2, 2, 10, 7, 4, 2, 2, 3, 1, 2, 1, 3, 3, 1, 1, 1, 23, 13, 2, 2, 1, 1, 2, 13, 9, 9, 1, 6, 1, 2, 1, 7, 2, 2, 12, 6, 1, 16, 8]]

[11, [294204, 55, 1, 1, 1, 2, 1, 76, 1, 10, 1, 1, 1, 1, 10, 1, 41, 3, 1, 1, 2, 2, 3, 6, 6, 2, 1, 5, 1, 1, 1, 12, 1, 61, 6, 2, 44, 1, 8, 1, 5, 5, 1, 2, 1, 5, 3, 1, 2, 8, 4, 1, 2, 3, 2, 2, 1, 1, 3, 1, 21, 2, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 1, 6]]

[12, [924269, 5, 1, 1, 27, 2, 20, 3, 1, 15, 3, 1, 1, 2, 2, 1, 5, 4, 3, 2, 1, 1, 6, 64, 9, 5, 3, 1, 8, 1, 1, 2, 1, 2, 8, 1, 2, 1, 1, 1, 2, 4, 1, 1, 26, 1, 50, 2, 1, 3, 1, 1, 19, 1, 4, 3, 6, 1, 1, 1, 15, 1, 3, 1, 1, 2, 1, 1, 7, 5, 1, 1, 1, 1, 5]]

[13, [2903677, 3, 1, 2, 3, 1, 1, 4, 1, 24, 1, 1, 1, 2, 1, 26, 1, 4, 2, 1, 5, 1, 1, 6, 1, 7, 22, 2, 1, 7, 5, 1, 4, 1, 1, 1, 1, 1, 6, 1, 1, 1, 5, 1, 2, 1, 2, 146, 2, 1, 1, 2, 2, 3, 5, 3, 2, 1, 1, 1, 24, 2, 2, 3, 2, 14, 1, 2, 1, 5, 11, 4, 3, 2, 1, 1, 5]]

[14, [9122171, 5, 1, 1, 128, 1, 14, 41, 2, 2, 1, 1, 1, 3, 1, 9, 4, 1, 1, 1, 144, 2, 1, 14, 3, 2, 10, 1, 1, 2044, 1, 2, 2, 2, 1, 3, 5, 1, 1, 1, 1, 1, 5, 404, 1, 1, 1, 1, 16, 1, 2, 21, 1, 1, 3, 46, 10, 12]]

[15, [28658145, 1, 31, 1, 2, 427, 1, 1, 6, 1, 1, 5, 1, 1, 4, 3, 14, 5, 7, 2, 1, 4, 1, 8, 6, 1, 2, 1, 1, 2, 1, 6, 15, 8, 21, 1, 1, 1, 11, 2, 2, 3, 1, 3, 2, 1, 3, 101, 1, 2, 3, 5, 1, 1, 10, 45, 4, 1, 4, 1, 3, 10, 3]]

[16, [90032220, 1, 5, 2, 1, 2, 1, 1, 1, 19, 1, 3, 1, 2, 7, 2, 2, 1, 1, 2, 8, 2, 1, 4, 17, 1, 7, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 2, 1, 12, 18, 1, 2, 4, 2, 5, 6, 1, 14, 1, 1, 1, 13, 1, 9, 1, 1, 33, 1, 1, 1, 2, 2, 2, 21, 3, 5, 3, 1, 1, 1, 4, 1, 2, 1, 1, 2, 4, 4]]

[17, [282844563, 1, 1, 2, 2, 1, 1, 3, 16, 1, 4, 2, 3, 2, 1, 2, 13, 2, 2, 6, 12, 5, 1, 3, 1, 1, 6, 1, 1, 1, 1, 3, 4, 3, 2, 2, 1, 3, 4, 2, 3, 1, 6, 15, 1, 2, 1, 1, 1, 13, 1, 1, 1, 7, 2, 20, 2, 1, 2, 2, 9, 3, 1, 2, 6, 1, 1, 3, 5, 1, 21, 1, 4, 3, 1, 4]]

[18, [888582403, 14, 30, 1, 4, 2, 1, 1, 2, 5, 152, 3, 1, 6, 5, 7, 2, 1, 9, 1, 1, 1, 267, 1, 1, 3, 37, 1, 5, 1, 1, 4, 1, 2, 2, 9, 1, 4, 4, 70, 1, 2, 1, 3, 1, 66, 4, 1, 2, 3, 1, 8, 2, 7, 1, 2, 9, 3, 6]]

[19, [2791563949, 1, 1, 2, 18, 6, 89, 163, 1, 68, 5, 1, 1, 2, 1, 4, 29, 13, 1, 1, 1, 1, 1, 3, 1, 2, 2, 70, 4, 1, 3, 6, 1, 2, 1, 4, 1, 3, 5, 5, 1, 2, 1, 1, 3, 3, 1, 3, 1, 6, 1, 1, 1, 1, 3, 5, 2, 1, 3, 2, 2, 1, 1, 1, 2, 6, 6]]

[20, [8769956796, 12, 10, 1, 6, 1, 5, 2, 1, 3, 1, 93, 1, 1, 2, 1, 12, 4, 3, 4, 5, 2, 2, 3, 6, 1, 1, 2, 14, 1, 14, 1, 1, 2, 2, 12, 44, 1, 1, 1, 6, 1, 1, 1, 4, 1, 3, 3, 4, 4, 2, 1, 4, 1, 9, 1, 3, 7, 6, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 2]]

[21, [27551631842, 1, 6, 1, 8, 3, 1, 7, 1, 1, 1, 2, 9, 1, 23, 1, 2, 649, 2, 2, 1, 8, 1, 2, 6, 1, 1, 1, 153, 2, 2, 743, 1, 22, 1, 2, 1, 11, 1, 3, 11, 2, 1, 1, 3, 1, 1, 1, 6, 1, 3, 4, 5, 1, 3, 2, 1, 2, 1, 16, 12]]

[22, [86556004191, 1, 52, 1, 1, 2, 7, 1, 1, 36, 1, 7, 2, 1, 57, 18, 11, 401, 2, 46, 2, 2, 1, 1, 1, 1, 3, 1, 2, 2, 1, 2, 19, 23, 2, 2, 1, 9, 4, 1, 1, 2, 2, 2, 7, 3, 3, 1, 2, 1, 34, 5, 1, 1, 4, 25]]

[23, [271923706893, 1, 1, 1, 1, 1, 1, 10, 4, 3, 2, 2, 3, 1, 26, 163, 1, 80, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 12, 2, 1, 1, 5, 1, 2, 10, 3, 2, 1, 5, 1, 16, 1, 51, 4, 1, 2, 2, 1, 8, 1, 1, 6, 2, 2, 1, 1, 1, 1, 1, 2, 3, 14, 1, 2, 1, 1, 29]]

[24, [854273519913, 1, 7, 1, 13, 2, 1, 4, 2, 1, 44, 1, 1, 4, 2, 1, 2, 4, 1, 1, 1, 6, 1, 4, 1, 1, 3, 1, 1, 1, 5, 3, 3, 31, 1, 1, 1, 5, 2, 1, 2, 1, 26, 8, 1, 2, 2, 8, 1, 7, 2, 1, 1, 1, 1, 1, 1, 1, 6, 2, 1, 23, 5, 2, 3, 1, 14, 1, 9]]

[25, [2683779414317, 1, 3, 4, 21, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 6, 1, 49, 1, 1, 13, 3, 1, 4, 1, 3, 7, 2, 8, 1, 5, 2, 1, 2, 3, 7, 1, 1, 2, 1, 1, 8, 1, 2, 1, 11, 1, 12, 1, 2, 1, 4, 9, 3, 1, 2, 19, 1, 1, 7, 1, 2, 2, 1, 1, 1, 4, 3, 2, 1, 5, 1, 1, 3]]

[26, [8431341691876, 4, 1, 4, 1, 6, 6, 7, 1, 2, 26, 1, 3, 1, 7, 4, 4, 1, 1, 1, 1, 1, 3, 1, 2, 4, 1, 1, 14, 65, 2, 3, 12, 1, 1, 3, 3, 2, 11, 5, 11, 1, 1, 4, 2, 2, 1, 9, 1, 1, 1, 1, 1, 18, 1, 1, 1, 2, 2, 3, 1, 2, 2, 1, 5, 1, 2, 2, 1, 1, 2]]

[27, [26487841119103, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 4, 1, 2, 2, 6, 2, 97, 1, 1, 7, 1, 1, 1, 1, 1, 3, 5, 1, 2, 1, 9, 1, 2, 1, 1, 5, 3, 57, 1, 4, 2, 11, 1, 1, 16, 6, 2, 2, 3, 1, 1, 1, 5, 20, 9, 4, 6, 9, 1, 1, 1, 3, 13, 1, 3, 7]]

[28, [83214007069229, 1, 1, 1, 1, 2, 1, 1, 1, 25, 1, 4, 1, 4, 1, 34, 17, 2, 1, 3, 1, 4, 1, 3, 5, 1, 5, 6, 4, 11, 3, 23, 1, 2, 3, 2, 5, 1, 1, 1, 6, 3, 3, 1, 1, 13, 1, 1, 10, 1, 16, 1, 1, 906, 1, 2, 27, 36]]

[29, [261424513284460, 1, 6, 1, 2, 1, 212, 1, 19, 1, 3, 1, 169, 2, 1, 1, 1, 26, 5, 6, 2, 10, 1, 3, 2, 4, 1, 8, 1, 3, 5, 21, 5, 2, 1, 1, 9, 1, 2, 1, 1, 1, 4, 2, 9, 1, 5, 18, 2, 2, 1, 1, 6, 1, 8, 1, 3, 3, 12]]

[30, [821289330402749, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 2, 10, 1, 2, 1, 1, 1, 3, 5, 1, 4, 1, 2, 2, 1, 2, 1, 2, 1, 6, 6, 2, 20, 11, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 4, 1, 3, 5, 6, 1, 1, 1, 2, 4, 2, 1, 8, 1, 6, 1, 1, 137, 7, 1, 3, 5, 1, 1, 2, 39, 3]]

[31, [2580156526864958, 1, 1, 23, 1, 1, 8, 12, 3, 4, 433, 1, 2, 13, 1, 1, 2, 1, 30, 2, 1, 1, 1, 5, 8, 11, 2, 1, 3, 7, 1, 9, 180, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 10, 2, 2, 1, 1, 1, 2, 2, 15, 649]]

[32, [8105800789910709, 1, 1, 1, 7, 1, 1, 3, 1, 8, 2, 2, 1, 3, 3, 2, 9, 2, 3, 99, 35, 5, 4, 3, 7, 1, 4, 1, 17, 1, 1, 1, 2, 2, 1, 1, 1, 10, 5, 70, 1, 40, 1, 2, 10, 7, 2, 2, 1, 1, 49, 1, 3, 16]]

[33, [25465124213045828, 2, 7, 1, 736, 3, 1, 5, 1, 35, 1, 12, 5, 3, 6, 3, 11, 2, 15, 1, 1, 2, 2, 2, 9, 2, 1, 4744, 4, 12, 3, 2, 15, 1, 1, 1, 200, 10, 1, 1, 3, 4, 6, 1, 2]]

[34, [80001047150456339, 1, 1, 4, 4, 1, 1, 16, 1, 1, 2, 2, 1, 1, 4, 37, 7, 1, 2, 1, 5, 1, 1, 2, 1, 24, 1, 8, 1, 1, 1, 1, 13, 2, 1, 13, 1, 3, 1, 3, 1, 2, 38, 3, 3, 1, 1, 2, 1, 2, 1, 4, 1, 1, 3, 2, 1, 124, 1, 11, 1, 2, 1, 12, 2, 2]]

[35, [251330702007364298, 1, 1, 1, 1, 1, 1, 7, 1, 3, 1, 1, 1, 5834, 1, 3, 34, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 1, 7, 63, 3, 6, 3, 1, 6, 1, 12, 6, 1, 2, 2, 15, 1, 6, 4, 2, 5, 1, 10, 1, 5, 18, 13, 1, 4]]

[36, [789578687047901181, 11, 3, 1, 6, 37, 6, 1, 6, 1, 2, 3, 3, 2, 2, 1, 3, 10, 2, 2, 3, 2, 1, 1, 4, 90, 1, 2, 1, 3, 1, 1, 28, 40, 1, 1, 3, 2, 5, 1, 4, 37, 1, 12, 3, 9, 5, 1, 4, 1, 1, 2, 1, 1, 6]]

[37, [2480534602660760780, 2, 3, 3, 1, 2, 5, 1, 1, 1, 5, 1, 2, 1, 64, 7, 1, 1, 7, 1, 1, 16, 1, 84, 1, 2, 1, 1, 3, 4, 1228, 13, 2, 1, 13, 1, 1, 2, 4, 1, 1, 10, 4, 5, 2, 3, 1, 3, 1, 2, 2, 1, 2, 1, 2, 7, 2]]

[38, [7792829284694322855, 1, 1, 1, 1, 1, 7, 3, 3, 141, 1, 2, 2, 2, 1, 1, 192, 1, 5, 1, 1, 3, 1, 8, 14, 1, 2, 49, 1, 12, 1, 4, 2, 1, 12, 2, 14, 12, 246, 1, 35, 1, 1, 3, 6, 1, 1, 1, 1, 14, 4]]

[39, [24481895231475088054, 1, 1, 1, 3, 1, 3, 1, 1, 7, 1, 1, 3, 42, 1, 150, 1, 2, 1, 1, 5, 1, 1, 1, 8, 2, 9, 1, 2, 1, 2, 2, 15, 2, 1, 1, 5, 1, 1, 1, 1, 4, 4, 2, 2, 1, 17, 3, 2, 1, 1, 2, 5, 24, 2, 8, 2, 1, 1, 1, 7, 1, 2]]

[40, [76912142205157127257, 3, 1, 3, 2, 1, 2, 1, 4, 1, 7, 79, 2, 1, 3, 2, 2, 1, 1, 5, 7, 31, 1, 1, 4, 1, 1, 15, 1, 2, 1, 1, 18, 1, 8, 1, 7, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 6, 4, 1, 7, 20, 1, 1, 3, 1, 1, 1, 3, 15, 2]]

[41, [241626620923575111130, 3, 2, 5, 4, 1, 2, 7, 1, 4, 8, 1, 18, 1, 5, 3, 1, 3, 1, 1, 2, 3, 4, 10, 1, 1, 10, 1, 2, 1, 4, 37, 1, 5, 1, 1, 2, 4, 1, 1, 1, 1, 1, 4, 1, 2, 3, 5, 1, 1, 110, 1, 1, 2, 3, 1, 4, 1, 9, 1, 1, 2]]

[42, [759092417205229390873, 1, 7, 11, 2, 1, 6, 2, 2, 1, 1, 2, 1, 2, 104, 25, 13, 6, 1, 3, 3, 2, 1, 7, 2, 2, 4, 2, 1, 5, 2, 3, 1, 1, 1, 1, 3, 1, 1, 1, 16, 5, 1, 18, 40, 2, 1, 2, 1, 1, 16, 4, 11]]

[43, [2384759161287667022584, 1, 16, 1, 2, 4, 1, 5, 1, 1, 12, 3, 1, 3, 178, 1, 23, 1, 5, 2, 1, 1, 4, 1, 1, 6, 1, 1, 2, 1, 4, 1, 14, 3, 1, 1, 10, 10, 1, 2, 12, 2, 1, 115, 1, 885, 1, 1, 2]]

[44, [7491941861682291566352, 1, 1, 4, 7, 5, 1, 1, 1, 5, 2, 1, 1, 5, 6, 1, 18, 2, 4, 139, 1, 25, 4, 2, 1, 1, 2, 86, 1, 3, 3, 1, 4, 3, 5, 1, 2, 10, 31, 2, 1, 3, 5, 14, 1, 2, 1, 1, 1, 1, 1, 1, 11]]

[45, [23536629513782926058032, 2, 23, 3, 4, 4, 2, 1, 2, 1, 1, 5, 1, 4, 2, 1, 1, 1, 1, 2, 2, 5, 1, 6, 2, 4, 2, 1, 5, 2, 30, 1, 1, 1, 3, 1, 39, 2, 5, 3, 4, 7, 1, 1, 40, 3, 12, 2, 1, 1, 2, 1, 8, 1, 1, 1, 15]]

[46, [73942502370765147684186, 1, 1, 2, 2, 8, 7, 1, 2, 2, 1, 16, 1, 1, 1, 1, 6, 1, 2, 1, 9, 20, 7, 1, 9, 10, 5, 9, 1, 1, 16, 1, 2, 1, 8, 1, 2, 1, 3, 10, 2, 1, 1, 1, 1, 4, 1, 1, 301, 1, 3, 1, 10]]

[47, [232297222236041657886385, 1, 3, 1, 77, 2, 13, 1, 1, 5, 3, 1, 3, 2, 24, 51, 25, 1, 1, 2, 6, 13, 4, 1, 1, 13, 3, 1, 1, 8, 1, 41, 3, 9, 13, 2, 7, 1, 1, 1, 14, 1, 1, 1, 5, 1, 4]]

[48, [729783246826064035206280, 1, 1, 12, 1, 2, 2, 2, 1, 10, 8, 2, 1, 1, 9, 5, 3, 8, 2, 1, 38, 3, 1, 4, 1, 20, 1, 1, 1, 16, 1, 2, 1, 7, 5, 1, 1, 10, 1, 1, 2, 6, 34, 4, 1, 1, 2, 3, 1, 2, 1, 13, 1, 1, 2, 2]]

[49, [2292681686941669566432911, 7, 4, 3, 1, 82, 5, 13, 3, 3, 1, 3, 1, 3, 1, 2, 1, 1, 5, 1, 11, 1, 110, 1, 11, 1, 1, 1, 3, 3, 1, 1, 1, 31, 1, 4, 1, 3, 7, 3, 14, 1, 2, 1, 4, 170, 3, 3]]

[50, [7202671944715803306364652, 1, 2, 19, 4, 1, 2, 1, 1, 116, 4, 1, 1, 1, 2, 2, 1, 7, 1, 3, 2, 8, 1, 2, 2, 9, 4, 9, 2, 3, 7, 1, 1, 1, 1, 7, 2, 1, 4, 11, 14, 1, 2, 2, 47, 3, 1, 1, 3, 2, 5]]

[51, [22627861267736477052126260, 8, 1, 4, 1, 1, 1, 3, 1, 1, 1, 215, 8, 170, 3, 2, 4, 1, 2, 7, 2, 1, 9, 1, 3, 1, 5, 2, 1, 3, 2, 3, 7, 1, 1, 1, 3, 3, 27, 1, 1, 1, 1, 9, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 3]]

[52, [71087522725169941823644668, 1, 4, 1, 2, 1, 1, 33, 2, 2, 2, 1, 1, 1, 8, 1, 2, 1, 40, 4, 4, 2, 1, 1, 3, 28, 1, 1, 1, 26, 1, 1, 29, 3, 2, 150, 1, 7, 3, 104, 4, 4, 2, 3, 1, 19]]

[53, [223328039155291367651964249, 2, 1, 2, 20, 1, 10, 1, 3, 1, 27, 5, 1, 1, 1, 6, 2, 3, 1, 2, 7, 3, 1, 1, 1, 3, 9, 1, 1, 2, 2, 1, 3, 2, 2, 1, 1, 1, 5, 3, 3, 3, 258, 4, 7, 1, 1, 2, 2, 2, 5, 1, 5, 2]]

[54, [701605727150877054142644067, 33, 1, 1, 1, 2, 3, 4, 8, 21, 1, 4, 3, 2, 9, 271, 34, 1, 2, 1, 6, 2, 2, 6, 1, 2, 3, 1, 43, 61, 1, 1, 2, 1, 21, 1, 39, 2, 2, 3]]

[55, [2204159398133720289741067354, 1, 1, 1, 1, 2, 8, 2, 1, 7, 1, 2, 1, 5, 19, 10, 3, 2, 3, 101, 5, 1, 6, 1, 1, 5, 2, 22, 4, 17, 1, 4, 3, 7, 2, 2, 6, 1, 1, 2, 5, 3, 1, 3, 4, 2]]

[56, [6924570972517795883320726281, 1, 1, 4, 1, 2, 10, 5, 1, 1, 3, 1, 5, 5, 5, 2, 12, 29, 1, 5, 1, 70, 8, 1, 3, 1, 1, 4, 3, 10, 2, 2, 17, 1, 5, 1, 10, 4, 2, 3, 1, 3, 1, 1, 1, 1, 3, 3]]

[57, [21754181296523037597639725717, 1, 6, 1, 1, 4, 3, 4, 1, 2, 3, 2, 3, 22, 1, 2, 2, 1, 1, 2, 2, 1, 4, 15, 1, 63, 6, 5, 3, 2, 17, 5, 1, 29, 1, 2, 3, 2, 3, 1, 1, 1, 4, 1, 2, 1, 2, 10]]

[58, [68342776146017258398967520977, 1, 188, 1, 2, 2, 1, 2, 2, 4, 3, 8, 1, 3, 3, 2, 1, 2, 3, 2, 4, 1, 3, 14, 1, 1, 2, 2, 5, 1, 1, 1, 1, 3, 1, 5, 1, 13, 1, 1, 1, 3, 14, 3, 15, 1, 4, 2, 4, 1, 1, 1, 1, 2]]

[59, [214705163466259581466085919504, 117, 1, 2, 1, 3, 1, 3, 4, 3, 9, 2, 1, 2, 4, 1, 1, 7, 3, 1, 2, 1, 2, 4, 2, 1, 2, 45, 1, 92, 2, 4, 3, 114, 5, 1, 29, 1, 1, 1, 1, 3, 11]]

[60, [674516164233396768200126958915, 1, 1, 3, 1, 2, 1, 1, 1, 131, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 8, 2, 1, 5, 1, 1, 17, 5, 10, 1, 3, 1, 1, 4, 2, 3, 1, 1, 1, 16, 1, 4, 2, 1, 2, 2, 2, 1, 3, 2, 2, 1, 13, 1, 25]]

[61, [2119055026283205737110078195749, 2, 36, 1, 1, 5, 1, 2, 1, 4, 2, 7, 1, 1, 9, 2, 2, 1, 71, 17, 1, 45, 4, 1, 1, 1, 26, 2, 3, 6, 3, 18, 2, 1, 3, 3, 1, 7, 1, 2, 1, 4, 3]]

[62, [6657207703123845367424584915447, 1, 1, 7, 2, 1, 3, 1, 8, 34, 1, 18, 3, 4, 2, 1, 1, 6, 3, 2, 1, 389, 1, 2, 21, 4, 7, 9, 1, 6, 1, 1, 1, 6, 1, 5, 6, 77]]

[63, [20914234813555253845753030071176, 1, 4, 1, 1, 1, 1, 1, 2, 2, 2, 1, 5, 1, 2, 2, 1, 1, 1, 1, 1, 71, 20, 1, 1, 1, 1, 6, 1, 1, 2, 1, 18, 6, 1, 10, 1, 6, 4, 1, 1, 2, 1, 52, 3, 3, 2, 5, 2, 7, 1, 2]]

[64, [65704006445717084572022626334540, 1, 31, 1, 1, 11, 1, 5, 11, 1, 47, 1, 5, 1, 1, 2, 28, 10, 1, 3, 2, 7, 1, 5, 1, 6, 5, 1, 11, 5, 3, 1, 5, 2, 1, 2, 2, 5, 2, 2, 1, 2, 1, 13]]

[65, [206415223961281214949635806621076, 8, 1, 2, 15, 2, 1, 3, 1, 2, 2, 1, 1, 1, 7, 1, 1, 7, 3, 3, 1, 1, 1, 5, 1, 20, 1, 3, 392, 1, 1, 2, 3, 2, 1, 4, 2, 1, 2, 17, 1, 25, 1, 6, 7]]

[66, [648472551185852924760805487074064, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 33, 1, 46, 1, 2, 3, 1, 1, 1, 4, 3, 2, 3, 4, 3, 1, 35, 1, 1, 1, 3, 22, 1, 3, 2, 14, 19, 1, 1, 1, 12, 8, 4]]

[67, [2037236602860106711985144771884927, 2, 1, 1, 5, 2, 1, 1, 1, 1, 1, 13, 1, 24, 1, 1, 1, 1, 5, 1, 4, 3, 2, 1, 1, 19, 1, 1, 2, 2, 1, 14, 1, 6, 2, 1, 2, 6, 1, 3, 2, 1, 6, 4, 3, 1, 7, 16, 3, 1, 1, 2]]

[68, [6400167545169538406684014293670341, 1, 1, 4, 10, 1, 6, 1, 1, 3, 1, 1, 4, 5, 1, 18, 1, 3, 1, 5, 1, 2, 7, 2, 4, 1, 1, 1, 1, 2, 6, 1, 2, 2, 1, 6, 4, 1, 1, 1, 18, 1, 4, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 17]]

[69, [20106719341648443041008873880038160, 1, 10, 11, 1, 54, 1, 2, 57, 2, 2, 1, 1, 10, 1, 1, 4, 9, 1, 1, 1, 1, 1, 1, 5, 1, 5, 4, 1, 5, 7, 6, 50, 1, 24, 1, 2, 6]]

[70, [63167121771514552681892760031018334, 1, 2, 1, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 2, 4, 51, 2, 2, 1, 2, 2, 1, 4, 1, 3, 2, 7, 3, 24, 539, 8, 1, 2, 2, 9, 1, 1, 5, 1, 1, 10]]

[71, [198445365705802004702001456166116205, 1, 3, 2, 34, 1, 4, 1, 5, 1, 1, 1, 10, 1, 1, 1, 1, 1, 10, 7, 36, 3, 6, 3, 1, 1, 2, 4, 4, 2, 1, 4, 11, 294, 1, 56, 2]]

[72, [623434503040287472342007784213127357, 1, 3, 1, 1, 2, 1, 6, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 5, 3, 4, 1, 1, 7, 44, 1, 1, 2, 13, 1, 97, 1, 3, 5, 1, 1, 1, 2, 1, 4, 2]]

[73, [1958577254745770740635072198655932631, 27, 1, 1, 1, 4, 1, 4, 35, 7, 1, 19, 1, 1, 5, 2, 4, 3, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 5, 8, 1, 1, 1, 1, 2, 41, 1, 1, 1, 48, 1, 19]]

[74, [6153051914997378363457293172896716008, 1, 18, 1, 4, 23, 2, 1, 15, 1, 2, 14, 1, 4, 3, 1, 13, 1, 3, 1, 1, 3, 4, 6, 1, 4, 2, 4, 2, 7, 4, 4, 5, 2, 9, 3, 2]]

[75, [19330382693312372796273669213182810875, 1, 3, 6, 1, 4, 2, 2, 11, 1, 2, 1, 1, 1, 1, 2, 3, 33, 1, 1, 1, 2, 2, 4, 1, 10, 1, 1, 5, 47, 13, 5987]]

[76, [60728188260389431620181363923786356434, 5, 16, 4, 3, 1, 3, 1, 4, 6, 1, 4, 12, 1, 5, 2, 2, 4, 1, 1, 7, 3, 1, 3, 3, 1, 1, 1, 2, 13, 1, 1, 1, 2, 1, 2, 1, 36, 9, 2, 4]]

[77, [190783230104657364116871823923953977404, 4, 4, 4, 15, 2, 2, 4, 1, 5, 1, 24, 1, 6, 2, 3, 3, 1, 11, 1, 4, 2, 2, 1, 3, 1, 2, 2, 1, 10, 1, 3, 6, 2, 36, 2, 4, 3]]

[78, [599363194124922655319699900093848879647, 1, 17, 1, 13, 1, 2, 6, 5, 4, 1, 8, 1, 1, 42, 2, 31, 1, 26, 7, 5, 1, 2, 1, 2, 1, 20, 1, 23, 5, 1, 5, 4]]

[79, [1882955007494970137423872697623397063368, 1, 8, 3, 2, 1, 3, 1, 2, 6, 2, 1, 6, 3, 3, 4, 1, 1, 1, 1, 3, 5, 2, 2, 2, 1, 4, 1, 4, 1, 1, 1, 1, 1, 18, 1, 6, 1, 3, 4, 1, 76]]

[80, [5915477618586312249935148194956040327282, 1, 2, 4, 1, 3, 3, 10, 1, 1, 12, 4, 7, 1, 25, 4, 10, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 3, 7, 1, 1, 1, 3, 1, 1, 1, 1, 27]]

[81, [18584021029025603512479467959285372526679, 270, 5, 3, 10, 4, 1, 2, 3, 1, 5, 1, 3, 1, 18, 2, 1, 1, 9, 1, 8, 953, 8, 3, 1, 8, 4]]

[82, [58383423938945065690063367398575102156630, 1, 1, 1, 50, 19, 6, 4, 1, 10, 1, 3, 1, 11, 2, 14, 5, 7, 6, 1, 1, 12, 1, 3, 1, 1, 10, 1, 1, 1, 84]]

[83, [183416935738008287619993279499823507509433, 1, 20, 3, 2, 6, 1, 4, 5, 3, 1, 1, 7, 1, 2, 1, 1, 1, 1, 24, 1, 1, 3, 39, 1, 8, 2, 4, 3, 6, 13, 1, 1, 10]]

[84, [576221297858478037757896202112843514258698, 1, 6, 3, 3, 3, 2, 2, 5, 5, 3, 1, 4, 1, 3, 1, 2, 2, 1, 1, 1, 1, 4, 38, 2, 2, 1, 110, 5, 1, 3, 1, 2, 1, 2, 2]]

[85, [1810252596194170662517162499694380571752661, 5, 2, 3, 1, 7, 1, 2, 1, 2, 1, 4, 1, 2, 2, 4, 1, 5, 2, 4, 1, 3, 1, 50, 1, 3, 48, 1, 7, 2, 1, 1, 1, 6, 1, 4, 5]]

[86, [5687076257345457055936827972911358112818998, 3, 59, 1, 9, 1, 2, 5, 1, 2, 1, 4, 1, 16, 1, 1, 1, 2, 1, 16, 1, 21, 1, 3, 1, 5, 1, 1, 1, 3, 1, 1, 16, 3, 1, 6]]

[87, [17866476990481424293061935394100969629693609, 1, 1, 1, 2, 1, 3, 2, 1, 13, 1, 6, 17, 13, 3, 9, 317, 1, 2, 1, 4, 1, 13, 1, 1, 4, 2, 5, 1, 1, 1, 1, 2]]

[88, [56129192858827520816193436882886842322337670, 25, 11, 2, 1, 2, 12, 9, 7, 1, 95, 99, 3, 1, 4, 228, 1, 5, 6, 5]]

[89, [176335059937217224018860024808747332179340048, 2, 51, 1, 6, 48, 1, 16, 1, 1, 1, 1, 2, 12, 1, 7, 3, 1, 72, 1, 1, 1, 2, 2, 1, 3, 19, 2, 1, 4, 1, 2]]

[90, [553972928869077498297581459358161277972163803, 10, 2, 1, 3, 8, 3, 2, 6, 17, 1, 17, 1, 1, 4, 2, 1, 1673, 1, 1, 1, 2, 2, 2, 1, 79, 3]]

[91, [1740357283622714955277630726239733032237686240, 9, 2, 2, 7, 3, 8, 6, 1, 8, 2, 3, 7, 2, 1, 2, 4, 5, 1, 1, 1, 39, 7, 1, 1, 1, 75, 2]]

[92, [5467493656850609485803318470436356848616987732, 1, 1, 1, 1, 4, 13, 22, 1, 1, 8, 1, 1, 3, 1, 2, 2, 1, 18, 12, 1, 2, 1, 5, 3, 1, 3, 1, 290, 8]]

[93, [17176637905910668669319694724112097956068639929, 16, 2, 6, 1, 14, 1, 12, 1, 6, 3, 6, 4, 1, 1, 1, 2, 1, 19, 1, 30, 1, 1, 1, 98, 1, 1, 1, 3, 2]]

[94, [53961999458580926862279289940854110180185778687, 1, 1, 1, 5, 2, 2, 3, 14, 2, 1, 12, 1, 31, 3, 3, 2, 2, 23, 1, 16, 1, 1, 8, 1, 3, 1, 2, 1, 5]]

[95, [169526621072094240051062813849433542430041251045, 1, 3, 1, 1, 1, 1, 1, 3, 1, 6, 2, 7, 1, 5, 1, 1, 35, 5, 1, 19, 12, 1, 2, 1, 4, 2, 33, 1, 1, 12]]

[96, [532583587347991902715777143640488633081518781241, 1, 2, 1, 1, 1, 2, 5, 1, 1, 3, 1, 2, 13, 1, 1, 1, 1, 6, 1, 52, 1, 2, 9, 3, 1, 1, 1, 16, 1, 1, 3, 1, 5, 10]]

[97, [1673160685434949314609400471476613579121899911697, 1, 11, 1, 1, 1, 1, 1, 40, 1, 1, 1, 4, 1, 2, 1, 14, 3, 3, 2, 1, 4, 1, 1, 9, 1, 36, 1, 2, 5, 1, 1, 4]]

[98, [5256389317637699735335333000834901305449119091054, 1, 2, 1, 19, 8, 1, 8, 1, 3, 2, 1, 2, 1, 2, 9, 1, 5, 12, 4, 1, 1, 2, 7, 1, 8, 1, 3, 3, 1, 3]]

[99, [16513434064698463682619509822252680822186405912861, 3, 3, 2, 9, 1, 1, 1, 6, 3, 1, 21, 1, 1, 24, 1, 68, 1, 11, 2, 36, 3, 1, 1, 2]]

[100, [51878483143196131920862615246303013562686760680405, 1, 3, 1, 1, 1, 6, 2, 2, 18, 1, 5, 11, 2, 1, 2, 1, 1, 3, 2, 16, 3, 1, 4, 3, 1, 3, 2, 4, 2]]

240325

PARI


シュワルツの定理

「ラマヌジャンの遺した関数」の2.7 に載っている定理を確かめてみる。

a(n, m) = binomial(n^2, n)%n^m;
for(m=1, 6, for(n=1, 50, print1(a(n, m),", ")); print)

出力結果

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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

0, 2, 3, 12, 5, 12, 7, 56, 9, 40, 11, 108, 13, 84, 90, 240, 17, 144, 19, 80, 315, 220, 23, 72, 25, 312, 27, 560, 29, 0, 31, 992, 759, 544, 770, 720, 37, 684, 1053, 520, 41, 252, 43, 1408, 1125, 1012, 47, 1872, 49, 1200, 

0, 6, 3, 28, 5, 120, 7, 120, 9, 440, 11, 828, 13, 1064, 315, 496, 17, 2088, 19, 2080, 7812, 3608, 23, 648, 25, 5720, 6588, 10752, 29, 1800, 31, 18400, 29073, 12104, 36295, 12384, 37, 16568, 36036, 54920, 41, 12600, 43, 69168, 74025, 28520, 47, 54864, 49, 36200, 

0, 6, 3, 28, 5, 1200, 7, 3192, 2925, 6440, 11, 828, 13, 14784, 10440, 12784, 17, 25416, 19, 34080, 118944, 205920, 23, 111240, 156275, 286936, 262467, 515648, 29, 244800, 31, 575456, 640002, 169320, 36295, 712224, 37, 1827344, 2230839, 2166920, 41, 2012976, 43, 1091376, 3445650, 1683232, 47, 54864, 823592, 2036200, 

0, 6, 84, 796, 5, 3792, 7, 27768, 22608, 56440, 11, 83772, 13, 168448, 61065, 471536, 17, 445320, 19, 994080, 2258235, 4188272, 23, 6083208, 156275, 743912, 13548492, 3588928, 29, 16444800, 31, 28887008, 13685133, 2841992, 42053795, 49421088, 37, 24763840, 9171162, 45686920, 41, 101587248, 43, 98541872, 15747525, 64367616, 47, 153998928, 58471602, 133286200, 

0, 6, 84, 1820, 6255, 34896, 16814, 126072, 258804, 456440, 805266, 2820924, 2227771, 3933216, 1579815, 7811568, 9939016, 26899272, 12380514, 994080, 47183346, 112414544, 32181738, 22008456, 205234400, 131439048, 157037562, 365006656, 184600370, 210844800, 429437296, 699975648, 796392993, 502631656, 1460144420, 2044804896, 901471478, 1767937536, 2264776137, 1888886920, 2780548865, 1016425872, 735042258, 6860107056, 6843288150, 9332701536, 1834760103, 3211646544, 9380154819, 8883286200, 

240225

PARI


Expansion of (1/x) * Series_Reversion( x * (1-x)^u * (1-x^s)^t ).

2通りで算出し、一致することを確認してみた。

a(n, s, t, u) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef(serreverse(x*(1-x)^u*(1-x^s)^t)/x, n);
diff(n, s, t, u) = a(n, s, t, u)-b(n, s, t, u);

for(s=1, 2, for(t=-5, 5, for(u=-5, 5, print1([s, t, u],": "); for(n=0, 10, if(diff(n, s, t, u)==0, print1(a(n, s, t, u),", "))); print)))

出力結果

[1, -5, -5]: 1, -10, 145, -2470, 46060, -910252, 18730855, -397089550, 8612835715, -190223180840, 4263421511271, 

[1, -5, -4]: 1, -9, 117, -1785, 29799, -527085, 9706503, -184138713, 3573805950, -70625252863, 1416298046436, 

[1, -5, -3]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 

[1, -5, -2]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 

[1, -5, -1]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -5, 0]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -5, 1]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -5, 2]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -5, 3]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -5, 4]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -5, 5]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -4, -5]: 1, -9, 117, -1785, 29799, -527085, 9706503, -184138713, 3573805950, -70625252863, 1416298046436, 

[1, -4, -4]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 

[1, -4, -3]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 

[1, -4, -2]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -4, -1]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -4, 0]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -4, 1]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -4, 2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -4, 3]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -4, 4]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -4, 5]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, -3, -5]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 

[1, -3, -4]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 

[1, -3, -3]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -3, -2]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -3, -1]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -3, 0]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -3, 1]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -3, 2]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -3, 3]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -3, 4]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, -3, 5]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, -2, -5]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 

[1, -2, -4]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -2, -3]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -2, -2]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -2, -1]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -2, 0]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -2, 1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -2, 2]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -2, 3]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, -2, 4]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, -2, 5]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, -1, -5]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -1, -4]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -1, -3]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -1, -2]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -1, -1]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -1, 0]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -1, 1]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -1, 2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, -1, 3]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, -1, 4]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, -1, 5]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 0, -5]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, 0, -4]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, 0, -3]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, 0, -2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, 0, -1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 0, 0]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 0, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 0, 2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 0, 3]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 0, 4]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 0, 5]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 1, -5]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, 1, -4]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, 1, -3]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, 1, -2]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 1, -1]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 1, 0]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 1, 1]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 1, 2]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 1, 3]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 1, 4]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 1, 5]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 2, -5]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, 2, -4]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, 2, -3]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 2, -2]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 2, -1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 2, 0]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 2, 1]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 2, 2]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 2, 3]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 2, 4]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 2, 5]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 

[1, 3, -5]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, 3, -4]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 3, -3]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 3, -2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 3, -1]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 3, 0]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 3, 1]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 3, 2]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 3, 3]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 3, 4]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 

[1, 3, 5]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 

[1, 4, -5]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 4, -4]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 4, -3]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 4, -2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 4, -1]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 4, 0]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 4, 1]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 4, 2]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 4, 3]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 

[1, 4, 4]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 

[1, 4, 5]: 1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 

[1, 5, -5]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 5, -4]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 5, -3]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 5, -2]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 5, -1]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 5, 0]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 5, 1]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 5, 2]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 

[1, 5, 3]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 

[1, 5, 4]: 1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 

[1, 5, 5]: 1, 10, 155, 2870, 58565, 1270752, 28765650, 671650110, 16057800980, 391139588190, 9672348219898, 

[2, -5, -5]: 1, -5, 30, -185, 1090, -5626, 19720, 45695, -1929840, 26595030, -282947496, 

[2, -5, -4]: 1, -4, 17, -60, 79, 1296, -17360, 142068, -873895, 3711532, -2373973, 

[2, -5, -3]: 1, -3, 7, 5, -192, 1347, -5258, 1557, 160270, -1442000, 6975729, 

[2, -5, -2]: 1, -2, 0, 26, -123, 98, 2224, -14430, 23967, 245364, -2067680, 

[2, -5, -1]: 1, -1, -4, 19, 11, -336, 676, 4699, -26764, -25371, 663520, 

[2, -5, 0]: 1, 0, -5, 0, 60, 0, -935, 0, 16555, 0, -316251, 

[2, -5, 1]: 1, 1, -3, -15, -1, 197, 527, -1711, -13530, -7993, 221780, 

[2, -5, 2]: 1, 2, 2, -10, -72, -222, -34, 3598, 20735, 50630, -109596, 

[2, -5, 3]: 1, 3, 10, 31, 72, -6, -1540, -13065, -78479, -381576, -1474818, 

[2, -5, 4]: 1, 4, 21, 124, 781, 5120, 34474, 236492, 1644222, 11543644, 81623504, 

[2, -5, 5]: 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, 2658968130, 

[2, -4, -5]: 1, -5, 31, -205, 1368, -8891, 54102, -286173, 1020910, 2638800, -110587783, 

[2, -4, -4]: 1, -4, 18, -76, 247, -92, -8136, 97092, -798813, 5301608, -28151584, 

[2, -4, -3]: 1, -3, 8, -7, -109, 1008, -5316, 16145, 19220, -665873, 5278504, 

[2, -4, -2]: 1, -2, 1, 18, -100, 196, 871, -8758, 29971, 27512, -858793, 

[2, -4, -1]: 1, -1, -3, 15, -1, -197, 527, 1711, -13530, 7993, 221780, 

[2, -4, 0]: 1, 0, -4, 0, 38, 0, -468, 0, 6545, 0, -98728, 

[2, -4, 1]: 1, 1, -2, -11, -8, 94, 336, -259, -5087, -10712, 41026, 

[2, -4, 2]: 1, 2, 3, -2, -39, -176, -442, -26, 6222, 36062, 113240, 

[2, -4, 3]: 1, 3, 11, 43, 170, 657, 2392, 7675, 17603, -11898, -529678, 

[2, -4, 4]: 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 

[2, -4, 5]: 1, 5, 36, 305, 2833, 27916, 286632, 3033513, 32858595, 362515725, 4059475368, 

[2, -3, -5]: 1, -5, 32, -225, 1651, -12336, 92460, -687161, 5006180, -35235655, 234112824, 

[2, -3, -4]: 1, -4, 19, -92, 420, -1624, 3545, 20372, -386621, 3865104, -31129539, 

[2, -3, -3]: 1, -3, 9, -19, -21, 561, -4093, 20493, -69030, 43819, 1571700, 

[2, -3, -2]: 1, -2, 2, 10, -72, 222, -34, -3598, 20735, -50630, -109596, 

[2, -3, -1]: 1, -1, -2, 11, -8, -94, 336, 259, -5087, 10712, 41026, 

[2, -3, 0]: 1, 0, -3, 0, 21, 0, -190, 0, 1950, 0, -21576, 

[2, -3, 1]: 1, 1, -1, -7, -10, 27, 152, 169, -949, -4286, -2646, 

[2, -3, 2]: 1, 2, 4, 6, -1, -58, -304, -1090, -2876, -4216, 9244, 

[2, -3, 3]: 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 8414640, 

[2, -3, 4]: 1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 

[2, -3, 5]: 1, 5, 37, 325, 3141, 32261, 345605, 3818501, 43197445, 497868805, 5825331205, 

[2, -2, -5]: 1, -5, 33, -245, 1939, -15961, 134843, -1159829, 10103730, -88820835, 785856036, 

[2, -2, -4]: 1, -4, 20, -108, 598, -3300, 17732, -90140, 411281, -1451980, 965192, 

[2, -2, -3]: 1, -3, 10, -31, 72, 6, -1540, 13065, -78479, 381576, -1474818, 

[2, -2, -2]: 1, -2, 3, 2, -39, 176, -442, 26, 6222, -36062, 113240, 

[2, -2, -1]: 1, -1, -1, 7, -10, -27, 152, -169, -949, 4286, -2646, 

[2, -2, 0]: 1, 0, -2, 0, 9, 0, -52, 0, 340, 0, -2394, 

[2, -2, 1]: 1, 1, 0, -3, -7, -4, 24, 85, 99, -215, -1196, 

[2, -2, 2]: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 

[2, -2, 3]: 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 

[2, -2, 4]: 1, 4, 24, 172, 1360, 11444, 100520, 911068, 8457504, 80006116, 768464312, 

[2, -2, 5]: 1, 5, 38, 345, 3454, 36786, 408848, 4687969, 55048310, 658645110, 8001060132, 

[2, -1, -5]: 1, -5, 34, -265, 2232, -19766, 181300, -1706737, 16392049, -159959240, 1581278838, 

[2, -1, -4]: 1, -4, 21, -124, 781, -5120, 34474, -236492, 1644222, -11543644, 81623504, 

[2, -1, -3]: 1, -3, 11, -43, 170, -657, 2392, -7675, 17603, 11898, -529678, 

[2, -1, -2]: 1, -2, 4, -6, -1, 58, -304, 1090, -2876, 4216, 9244, 

[2, -1, -1]: 1, -1, 0, 3, -7, 4, 24, -85, 99, 215, -1196, 

[2, -1, 0]: 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42, 

[2, -1, 1]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[2, -1, 2]: 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 

[2, -1, 3]: 1, 3, 14, 79, 494, 3294, 22952, 165127, 1217270, 9146746, 69799476, 

[2, -1, 4]: 1, 4, 25, 188, 1563, 13840, 127972, 1221260, 11938471, 118936100, 1203155633, 

[2, -1, 5]: 1, 5, 39, 365, 3772, 41491, 476410, 5644477, 68493324, 846937140, 10633195119, 

[2, 0, -5]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[2, 0, -4]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[2, 0, -3]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[2, 0, -2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[2, 0, -1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[2, 0, 0]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[2, 0, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[2, 0, 2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[2, 0, 3]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[2, 0, 4]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[2, 0, 5]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[2, 1, -5]: 1, -5, 36, -305, 2833, -27916, 286632, -3033513, 32858595, -362515725, 4059475368, 

[2, 1, -4]: 1, -4, 23, -156, 1162, -9192, 75819, -644908, 5616182, -49826712, 448771622, 

[2, 1, -3]: 1, -3, 13, -67, 381, -2307, 14589, -95235, 636925, -4341763, 30056445, 

[2, 1, -2]: 1, -2, 6, -22, 90, -394, 1806, -8558, 41586, -206098, 1037718, 

[2, 1, -1]: 1, -1, 2, -5, 14, -42, 132, -429, 1430, -4862, 16796, 

[2, 1, 0]: 1, 0, 1, 0, 3, 0, 12, 0, 55, 0, 273, 

[2, 1, 1]: 1, 1, 3, 9, 32, 119, 466, 1881, 7788, 32868, 140907, 

[2, 1, 2]: 1, 2, 8, 38, 201, 1134, 6688, 40734, 254237, 1617572, 10452416, 

[2, 1, 3]: 1, 3, 16, 103, 735, 5592, 44452, 364815, 3067558, 26290517, 228819168, 

[2, 1, 4]: 1, 4, 27, 220, 1984, 19064, 191325, 1981932, 21031965, 227463808, 2498039219, 

[2, 1, 5]: 1, 5, 41, 405, 4423, 51441, 624687, 7828853, 100497980, 1314688375, 17462945394, 

[2, 2, -5]: 1, -5, 37, -325, 3141, -32261, 345605, -3818501, 43197445, -497868805, 5825331205, 

[2, 2, -4]: 1, -4, 24, -172, 1360, -11444, 100520, -911068, 8457504, -80006116, 768464312, 

[2, 2, -3]: 1, -3, 14, -79, 494, -3294, 22952, -165127, 1217270, -9146746, 69799476, 

[2, 2, -2]: 1, -2, 7, -30, 143, -728, 3876, -21318, 120175, -690690, 4032015, 

[2, 2, -1]: 1, -1, 3, -9, 32, -119, 466, -1881, 7788, -32868, 140907, 

[2, 2, 0]: 1, 0, 2, 0, 11, 0, 80, 0, 665, 0, 5980, 

[2, 2, 1]: 1, 1, 4, 13, 55, 232, 1052, 4869, 23206, 112519, 554560, 

[2, 2, 2]: 1, 2, 9, 46, 264, 1612, 10291, 67830, 458109, 3153744, 22049065, 

[2, 2, 3]: 1, 3, 17, 115, 863, 6903, 57687, 497683, 4398980, 39630305, 362562226, 

[2, 2, 4]: 1, 4, 28, 236, 2202, 21892, 227324, 2436508, 26752627, 299369532, 3401762928, 

[2, 2, 5]: 1, 5, 42, 425, 4756, 56686, 705500, 9061841, 119225535, 1598582580, 21763800786, 

[2, 3, -5]: 1, -5, 38, -345, 3454, -36786, 408848, -4687969, 55048310, -658645110, 8001060132, 

[2, 3, -4]: 1, -4, 25, -188, 1563, -13840, 127972, -1221260, 11938471, -118936100, 1203155633, 

[2, 3, -3]: 1, -3, 15, -91, 612, -4389, 32890, -254475, 2017356, -16301164, 133767543, 

[2, 3, -2]: 1, -2, 8, -38, 201, -1134, 6688, -40734, 254237, -1617572, 10452416, 

[2, 3, -1]: 1, -1, 4, -13, 55, -232, 1052, -4869, 23206, -112519, 554560, 

[2, 3, 0]: 1, 0, 3, 0, 24, 0, 253, 0, 3045, 0, 39627, 

[2, 3, 1]: 1, 1, 5, 17, 83, 381, 1939, 9905, 52544, 282315, 1545130, 

[2, 3, 2]: 1, 2, 10, 54, 332, 2162, 14734, 103630, 746857, 5486206, 40926152, 

[2, 3, 3]: 1, 3, 18, 127, 996, 8322, 72644, 654615, 6043455, 56866028, 543368586, 

[2, 3, 4]: 1, 4, 29, 252, 2425, 24864, 266270, 2943308, 33329204, 384641836, 4507577438, 

[2, 3, 5]: 1, 5, 43, 445, 5094, 62111, 790828, 10392109, 139883065, 1918862510, 26727571728, 

[2, 4, -5]: 1, -5, 39, -365, 3772, -41491, 476410, -5644477, 68493324, -846937140, 10633195119, 

[2, 4, -4]: 1, -4, 26, -204, 1771, -16380, 158224, -1577532, 16112057, -167710664, 1772645420, 

[2, 4, -3]: 1, -3, 16, -103, 735, -5592, 44452, -364815, 3067558, -26290517, 228819168, 

[2, 4, -2]: 1, -2, 9, -46, 264, -1612, 10291, -67830, 458109, -3153744, 22049065, 

[2, 4, -1]: 1, -1, 5, -17, 83, -381, 1939, -9905, 52544, -282315, 1545130, 

[2, 4, 0]: 1, 0, 4, 0, 42, 0, 580, 0, 9139, 0, 155664, 

[2, 4, 1]: 1, 1, 6, 21, 116, 566, 3176, 17501, 101391, 590756, 3519782, 

[2, 4, 2]: 1, 2, 11, 62, 405, 2784, 20066, 149158, 1136276, 8821958, 69556554, 

[2, 4, 3]: 1, 3, 19, 139, 1134, 9849, 89372, 837147, 8033545, 78573186, 780388752, 

[2, 4, 4]: 1, 4, 30, 268, 2653, 27980, 308212, 3504380, 40817586, 484534720, 5840593274, 

[2, 4, 5]: 1, 5, 44, 465, 5437, 67716, 880720, 11822217, 162556349, 2277870665, 32411053200, 

[2, 5, -5]: 1, -5, 40, -385, 4095, -46376, 548340, -6690585, 83615350, -1064887395, 13770292256, 

[2, 5, -4]: 1, -4, 27, -220, 1984, -19064, 191325, -1981932, 21031965, -227463808, 2498039219, 

[2, 5, -3]: 1, -3, 17, -115, 863, -6903, 57687, -497683, 4398980, -39630305, 362562226, 

[2, 5, -2]: 1, -2, 10, -54, 332, -2162, 14734, -103630, 746857, -5486206, 40926152, 

[2, 5, -1]: 1, -1, 6, -21, 116, -566, 3176, -17501, 101391, -590756, 3519782, 

[2, 5, 0]: 1, 0, 5, 0, 65, 0, 1110, 0, 21620, 0, 455126, 

[2, 5, 1]: 1, 1, 7, 25, 154, 787, 4812, 28169, 176065, 1096342, 7020112, 

[2, 5, 2]: 1, 2, 12, 70, 483, 3478, 26336, 205438, 1642890, 13388000, 110796476, 

[2, 5, 3]: 1, 3, 20, 151, 1277, 11484, 107920, 1046815, 10402541, 105357279, 1083566504, 

[2, 5, 4]: 1, 4, 31, 284, 2886, 31240, 353199, 4121772, 49274392, 600342184, 7427285236, 

[2, 5, 5]: 1, 5, 45, 485, 5785, 73501, 975225, 13354725, 187331895, 2677999545, 38873136507, 

240106

PARI


A292915

Fubini number に関係する数列を出力してみた。

N=15;

a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n);
for(n=0, N, for(k=0, n, print1(A(k, n-k), ", ")))

\\ 別の計算方法と一致することを確認
B(n, k) = k^n+sum(j=1, n, binomial(n, j)*B(n-j, k));
for(n=0, N, for(k=0, n, print1(A(k, n-k)-B(k, n-k), ", ")))

出力結果

1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563, 1, 11, 102, 789, 5094, 28207, 144038, 754241, 4366422, 28349043, 204495126, 1622632573, 1, 12, 123, 1066, 7787, 48638, 272451, 1492098, 8726283, 56697574, 408990251, 3245265146, 28091567595, 1, 13, 146, 1403, 11478, 80469, 498246, 2906071, 17387030, 113375465, 817979478, 6490530291, 56183135190, 526858348381, 1, 14, 171, 1806, 16395, 128170, 878843, 5532206, 34383435, 226488786, 1635899907, 12981058534, 112366270379, 1053716696762, 10641342970443, 1, 15, 198, 2281, 22790, 197291, 1495542, 10240869, 67087254, 451024447, 3270751238, 25961939921, 224732536662, 2107433393523, 21282685940886, 230283190977853, 

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, 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, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

231223

SINGULAR

n次多項式の判別式の項数について

この記事は
日曜数学 Advent Calendar 2023
の12/23 分として書いております。

2013年にn=17について計算されて以降、新しい情報はないようですが、

現段階で、普通のPCでどの程度計算可能か確認してみた。

一般にn次多項式f(x)（=a_0*x^n + a_1*x^(n-1) + ... +a_n）の判別式は、

(-1)^(n*(n-1)/2) * 1/a_0 * resultant(f(x), f'(x))

と表される。

判別式の項数を考えているので、a_0=1としても良い。

様々な計算方法があるが、

http://syskiso.fuee.u-fukui.ac.jp/~kkimur/FINAL.pdf

のP.20-21のアルゴリズムを使うことにする。

このアルゴリズムを使うと、

x^3 + a_1*x^2 + a_2*x + a_3 = 0 の判別式は次のように計算できる。

① x^2 + a_1*x + a_2 = 0 の判別式はa_1^2 - 4*a_2.

② D0 = a_2^2*(a_1^2 - 4*a_2) = a_1^2*a_2^2 - 4*a_2^3.

③ D1 = -4*a_1^3 + 18*a_1*a_2, D2 = -27.

④ D = D0 + D1*a_3 + D_2*a_3^2.

SINGULARを使うと、n=14までは現実的な時間で算出することができる。

ring r=(0),(a14,a13,a12,a11,a10,a9,a8,a7,a6,a5,a4,a3,a2,a1),lp;
list v=1,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14;
poly s=v[2]^2-4*v[3]; // 2次の場合
poly t0;
poly u0;
int n,j,k,w;
for(n=3; n<15; n++){
  t0=v[n]^2*s; // D0=a{n-1}^2*disc.(f{n-1}(x))
  s=t0;
  for(k=1; k<n; k++){
    u0=n*diff(t0,v[2]);
    for(j=2; j<n; j++){
      u0=u0+(n+1-j)*v[j]*diff(t0,v[j+1]);
    }
    t0=-u0/(k*v[n]);
    s=s+t0*v[n+1]^k; // D=D0+D1*an+D2*an^2+...+D{n-1}*an^(n-1)
  }
  print(size(s));
}

quit;

出力結果

5

16

59

246

1103

5247

26059

133881

706799

3815311

20979619

117178725

Auf Wiedersehen.

231112

PARI


A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s.

A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s

を満たすA(x)を展開してみた。

また、

A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s

を満たすA(x)も展開してみた。

K=5;
L=3;
M=30;
N=10;

a(n, s, t, u) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));
b(n, s, t, u) = my(A=1); for(i=1, n, A = 1 + x*A^t / (1 - x*A^u)^s + x*O(x^n) ); polcoeff(A, n);

\\ aとbの値が一致するかどうかを確認
for(s=0, K, for(t=0, K, for(u=0, K, for(n=0, M, if(a(n, s, t, u)!=b(n, s, t, u), print([n, s, t, u]))))))
for(s=0, L, for(t=0, L, for(u=0, L, print1([s, t, u],": "); for(n=0, N, print1(a(n, s, t, u),", ")); print)))

print

a(n, s, t, u) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));
b(n, s, t, u) = my(A=1); for(i=1, n, A = 1 + x*A^t * (1 + x*A^u)^s + x*O(x^n) ); polcoeff(A, n);

\\ aとbの値が一致するかどうかを確認
for(s=0, K, for(t=0, K, for(u=0, K, for(n=0, M, if(a(n, s, t, u)!=b(n, s, t, u), print([n, s, t, u]))))))
for(s=0, L, for(t=0, L, for(u=0, L, print1([s, t, u],": "); for(n=0, N, print1(a(n, s, t, u),", ")); print)))

出力結果

[0, 0, 0]: 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[0, 0, 1]: 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[0, 0, 2]: 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[0, 0, 3]: 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[0, 1, 0]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[0, 1, 1]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[0, 1, 2]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[0, 1, 3]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[0, 2, 0]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[0, 2, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[0, 2, 2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[0, 2, 3]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[0, 3, 0]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[0, 3, 1]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[0, 3, 2]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[0, 3, 3]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[1, 0, 0]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[1, 0, 1]: 1, 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 

[1, 0, 2]: 1, 1, 1, 3, 8, 25, 81, 274, 953, 3389, 12265, 

[1, 0, 3]: 1, 1, 1, 4, 13, 50, 201, 841, 3627, 15993, 71803, 

[1, 1, 0]: 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 

[1, 1, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 1, 2]: 1, 1, 2, 6, 21, 80, 322, 1347, 5798, 25512, 114236, 

[1, 1, 3]: 1, 1, 2, 7, 29, 131, 627, 3124, 16032, 84162, 449828, 

[1, 2, 0]: 1, 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, 171369, 

[1, 2, 1]: 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 

[1, 2, 2]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[1, 2, 3]: 1, 1, 3, 13, 66, 366, 2148, 13115, 82449, 530095, 3469401, 

[1, 3, 0]: 1, 1, 4, 19, 101, 578, 3479, 21714, 139269, 912354, 6078832, 

[1, 3, 1]: 1, 1, 4, 20, 113, 688, 4404, 29219, 199140, 1385904, 9807820, 

[1, 3, 2]: 1, 1, 4, 21, 126, 818, 5594, 39693, 289510, 2157150, 16348960, 

[1, 3, 3]: 1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 

[2, 0, 0]: 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 

[2, 0, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[2, 0, 2]: 1, 1, 2, 7, 26, 107, 462, 2074, 9572, 45147, 216638, 

[2, 0, 3]: 1, 1, 2, 9, 40, 202, 1068, 5884, 33356, 193365, 1140940, 

[2, 1, 0]: 1, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 

[2, 1, 1]: 1, 1, 3, 10, 37, 146, 602, 2563, 11181, 49720, 224540, 

[2, 1, 2]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[2, 1, 3]: 1, 1, 3, 14, 75, 438, 2704, 17356, 114661, 774514, 5324812, 

[2, 2, 0]: 1, 1, 4, 16, 68, 304, 1412, 6752, 33028, 164512, 831620, 

[2, 2, 1]: 1, 1, 4, 18, 90, 481, 2690, 15547, 92124, 556664, 3417062, 

[2, 2, 2]: 1, 1, 4, 20, 114, 702, 4550, 30585, 211270, 1490561, 10695354, 

[2, 2, 3]: 1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 

[2, 3, 0]: 1, 1, 5, 27, 161, 1030, 6921, 48190, 344669, 2517303, 18695908, 

[2, 3, 1]: 1, 1, 5, 29, 189, 1325, 9757, 74429, 583037, 4662653, 37911037, 

[2, 3, 2]: 1, 1, 5, 31, 219, 1672, 13439, 112043, 960017, 8402085, 74791408, 

[2, 3, 3]: 1, 1, 5, 33, 251, 2073, 18069, 163600, 1523731, 14504988, 140499307, 

[3, 0, 0]: 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 

[3, 0, 1]: 1, 1, 3, 9, 31, 114, 438, 1739, 7077, 29364, 123756, 

[3, 0, 2]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[3, 0, 3]: 1, 1, 3, 15, 82, 495, 3147, 20812, 141621, 985287, 6976369, 

[3, 1, 0]: 1, 1, 4, 13, 41, 129, 406, 1278, 4023, 12664, 39865, 

[3, 1, 1]: 1, 1, 4, 16, 71, 336, 1660, 8464, 44207, 235306, 1271807, 

[3, 1, 2]: 1, 1, 4, 19, 104, 615, 3829, 24728, 164122, 1112641, 7671781, 

[3, 1, 3]: 1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 

[3, 2, 0]: 1, 1, 5, 23, 111, 562, 2952, 15948, 88076, 495077, 2823293, 

[3, 2, 1]: 1, 1, 5, 26, 150, 925, 5967, 39772, 271758, 1893431, 13400897, 

[3, 2, 2]: 1, 1, 5, 29, 192, 1372, 10314, 80390, 643774, 5264984, 43788393, 

[3, 2, 3]: 1, 1, 5, 32, 237, 1906, 16179, 142665, 1294115, 11998349, 113194205, 

[3, 3, 0]: 1, 1, 6, 36, 236, 1656, 12192, 92960, 727824, 5817696, 47281472, 

[3, 3, 1]: 1, 1, 6, 39, 284, 2223, 18267, 155445, 1358073, 12111306, 109802183, 

[3, 3, 2]: 1, 1, 6, 42, 335, 2886, 26166, 246028, 2377161, 23459250, 235452723, 

[3, 3, 3]: 1, 1, 6, 45, 389, 3648, 36090, 370793, 3918225, 42314583, 464953943, 

[0, 0, 0]: 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[0, 0, 1]: 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[0, 0, 2]: 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[0, 0, 3]: 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[0, 1, 0]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[0, 1, 1]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[0, 1, 2]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[0, 1, 3]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[0, 2, 0]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[0, 2, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[0, 2, 2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[0, 2, 3]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[0, 3, 0]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[0, 3, 1]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[0, 3, 2]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[0, 3, 3]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[1, 0, 0]: 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 0, 1]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[1, 0, 2]: 1, 1, 1, 2, 3, 6, 11, 22, 44, 90, 187, 

[1, 0, 3]: 1, 1, 1, 3, 6, 16, 42, 114, 322, 918, 2673, 

[1, 1, 0]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 

[1, 1, 1]: 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 

[1, 1, 2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 1, 3]: 1, 1, 2, 6, 20, 72, 273, 1073, 4333, 17869, 74937, 

[1, 2, 0]: 1, 1, 3, 9, 31, 113, 431, 1697, 6847, 28161, 117631, 

[1, 2, 1]: 1, 1, 3, 10, 38, 154, 654, 2871, 12925, 59345, 276835, 

[1, 2, 2]: 1, 1, 3, 11, 46, 207, 979, 4797, 24138, 123998, 647615, 

[1, 2, 3]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[1, 3, 0]: 1, 1, 4, 18, 94, 529, 3135, 19270, 121732, 785496, 5155167, 

[1, 3, 1]: 1, 1, 4, 19, 104, 614, 3816, 24595, 162896, 1101922, 7580904, 

[1, 3, 2]: 1, 1, 4, 20, 115, 715, 4683, 31824, 222300, 1586310, 11514030, 

[1, 3, 3]: 1, 1, 4, 21, 127, 833, 5763, 41401, 305877, 2309385, 17739561, 

[2, 0, 0]: 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 

[2, 0, 1]: 1, 1, 2, 3, 6, 11, 22, 44, 90, 187, 392, 

[2, 0, 2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[2, 0, 3]: 1, 1, 2, 7, 24, 95, 386, 1641, 7150, 31844, 144216, 

[2, 1, 0]: 1, 1, 3, 6, 13, 28, 60, 129, 277, 595, 1278, 

[2, 1, 1]: 1, 1, 3, 8, 25, 81, 274, 953, 3389, 12265, 45025, 

[2, 1, 2]: 1, 1, 3, 10, 39, 162, 708, 3202, 14867, 70448, 339324, 

[2, 1, 3]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[2, 2, 0]: 1, 1, 4, 14, 56, 237, 1046, 4762, 22198, 105430, 508384, 

[2, 2, 1]: 1, 1, 4, 16, 74, 364, 1876, 9993, 54582, 304040, 1720576, 

[2, 2, 2]: 1, 1, 4, 18, 94, 527, 3108, 18993, 119214, 763997, 4978304, 

[2, 2, 3]: 1, 1, 4, 20, 116, 728, 4818, 33100, 233824, 1687764, 12393520, 

[2, 3, 0]: 1, 1, 5, 25, 145, 905, 5941, 40433, 282721, 2018897, 14661349, 

[2, 3, 1]: 1, 1, 5, 27, 169, 1138, 8061, 59188, 446455, 3438863, 26935372, 

[2, 3, 2]: 1, 1, 5, 29, 195, 1415, 10815, 85748, 698763, 5816595, 49246703, 

[2, 3, 3]: 1, 1, 5, 31, 223, 1738, 14289, 121985, 1070995, 9609132, 87716242, 

[3, 0, 0]: 1, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 

[3, 0, 1]: 1, 1, 3, 6, 16, 42, 114, 322, 918, 2673, 7875, 

[3, 0, 2]: 1, 1, 3, 9, 34, 132, 546, 2327, 10191, 45534, 206788, 

[3, 0, 3]: 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 

[3, 1, 0]: 1, 1, 4, 10, 26, 69, 181, 476, 1252, 3292, 8657, 

[3, 1, 1]: 1, 1, 4, 13, 50, 201, 841, 3627, 15993, 71803, 327082, 

[3, 1, 2]: 1, 1, 4, 16, 77, 393, 2113, 11761, 67217, 392140, 2325691, 

[3, 1, 3]: 1, 1, 4, 19, 107, 648, 4144, 27500, 187654, 1308361, 9280049, 

[3, 2, 0]: 1, 1, 5, 20, 90, 430, 2136, 10937, 57307, 305822, 1656482, 

[3, 2, 1]: 1, 1, 5, 23, 123, 700, 4170, 25677, 162101, 1043603, 6825429, 

[3, 2, 2]: 1, 1, 5, 26, 159, 1042, 7185, 51340, 376806, 2823734, 21516113, 

[3, 2, 3]: 1, 1, 5, 29, 198, 1459, 11343, 91541, 759664, 6441671, 55570406, 

[3, 3, 0]: 1, 1, 6, 33, 209, 1425, 10206, 75751, 577494, 4495368, 35582439, 

[3, 3, 1]: 1, 1, 6, 36, 251, 1881, 14817, 120950, 1014042, 8680377, 75552553, 

[3, 3, 2]: 1, 1, 6, 39, 296, 2421, 20853, 186300, 1710522, 16041834, 153014400, 

[3, 3, 3]: 1, 1, 6, 42, 344, 3048, 28491, 276517, 2759943, 28150473, 292122423, 

231001

Ruby


Rational function

f, g を多項式とする。

f/g の計算は、1/gと大差がない。 

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 Rational_function(ary0, ary, n)
  a = ary0.clone
  i = 0
  while i < n
    a[i + 1] -= (0..i).inject(0){|s, j| s + ary[1 + i - j] * a[j]}
    i += 1
  end
  a
end

a = [1, -2] + [0] * 9

p I(a, 10)

# a/a = 1
p Rational_function(a, a, 10)
p a

b = [0, 0, 1] + [0] * 8
p Rational_function(b, a, 10)
p b

出力結果

[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024]

[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

[1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0]

[0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256]

[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]

230908

PARI


A357782, A357783 and A357784

出力してみた。

a(n) = if(n==0, 1, 2 * sum(k=0, n-1, binomial(n-1,k)*c(k)));
b(n) = if(n==0, 0,     sum(k=0, n-1, binomial(n-1,k)*a(k)));
c(n) = if(n==0, 0,     sum(k=0, n-1, binomial(n-1,k)*b(k)));

for(n=0, 20, print1(a(n),", "))
for(n=0, 20, print1(b(n),", "))
for(n=0, 20, print1(c(n),", "))

出力結果

1, 0, 0, 2, 12, 50, 184, 686, 2996, 16642, 110328, 784190, 5645876, 40685762, 296458344, 2226254766, 17564381332, 147289101090, 1312394060536, 12305546886398, 119906479624084, 

0, 1, 1, 1, 3, 21, 131, 705, 3515, 17389, 91739, 547889, 3746227, 28241373, 224124083, 1821051233, 15023818091, 126366334125, 1094358852075, 9858890038513, 92983173940419, 

0, 0, 1, 3, 7, 17, 61, 343, 2231, 14301, 88561, 542011, 3397483, 22638993, 164336085, 1299899087, 10991061663, 97070035205, 881323166809, 8173386231395, 77489746906355, 

230814

Ruby


素数を生成する漸化式(2)

素数だけを出力することができる。

require 'prime'

def A020639(n)
  return 1 if n == 1
  Prime.each(n){|i|
    return i if n % i == 0
  }
end

def A137613(n)
  ary = []
  a, b = 5, 5
  while ary.size < n
    ary << b
    a += b - 1
    b = A020639(a)
  end
  ary
end

n = 100
p A137613(n)

出力結果

[5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3, 941, 3, 7, 1889, 3, 3779, 3, 7559, 3, 13, 15131, 3, 53, 3, 7, 30323, 3, 60647, 3, 5, 3, 101, 3, 121403, 3, 242807, 3, 5, 3, 19, 7, 5, 3, 47, 3, 37, 5, 3, 17, 3, 199, 53, 3, 29, 3, 486041, 3, 7, 421, 23, 3, 972533, 3, 577, 7, 1945649, 3, 163, 7, 3891467, 3, 5, 3, 127, 443, 3, 31, 7783541, 3, 7, 15567089, 3, 19, 29, 3, 5323, 7, 5]