221002

PARI


A357392, A357393等

出力してみた。

A(n) = my(N=15, x='x+O('x^N)); sum(k=1, N, binomial(n*k, k)*x^k/(n*k));
B(n) = exp(A(n));
C(n) = B(n)-x*exp(n*A(n));

for(n=1, 5, print(Vec(serlaplace(A(n)))))
for(n=1, 5, print(Vec(B(n))))
for(n=1, 5, print(C(n)))

出力結果

[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800]

[1, 3, 20, 210, 3024, 55440, 1235520, 32432400, 980179200, 33522128640, 1279935820800, 53970627110400, 2490952020480000, 124903451312640000]

[1, 5, 56, 990, 24024, 742560, 27907200, 1235591280, 62990928000, 3634245014400, 234102016512000, 16654322805120000, 1296884927852236800, 109720581991308288000]

[1, 7, 110, 2730, 93024, 4037880, 213127200, 13253058000, 948964262400, 76899763100160, 6957624460550400, 695236239163065600, 76043127767523840000, 9036546669251861760000]

[1, 9, 182, 5814, 255024, 14250600, 968330880, 77519922480, 7146019520640, 745520860465920, 86839771951296000, 11171585428619616000, 1573144097507348889600, 240679032790296606105600]

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

[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440]

[1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 8414640, 50067108, 300830572, 1822766520]

[1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 1882933364, 15875338990, 134993766600]

[1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, 2658968130, 28558343775, 309831575760, 3390416787880]

1 + O(x^15)

1 + O(x^15)

1 + O(x^15)

1 + O(x^15)

1 + O(x^15)