A381745 Expansion of exp( Sum_{k>=1} binomial(8*k-1,2*k) * x^k/k ).

1, 21, 903, 49525, 3070308, 204928371, 14369906538, 1043861319189, 77866470852108, 5929621690613108, 459076176165983247, 36026517938705145267, 2859318461620989381900, 229114879928544260792946, 18509862380800289696106372, 1506048000721264678984095445, 123303480420582227597300406588

Offset: 0

Author

Seiichi Manyama, Mar 05 2025

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..514

Crossrefs

Cf. A079489, A381744, A381746.

Cf. A006632, A381751.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(8*k-1, 2*k)*x^k/k)))

Formula

G.f. A(x) satisfies A(x^2) = B(x)/x * B(-x)/(-x), where B(x) is the g.f. of A006632.

a(n) = Sum_{k=0..2*n} (-1)^k * A006632(k+1) * A006632(2*n-k+1).

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(8*k-1,2*k) * a(n-k).

G.f.: B(x)^3, where B(x) is the g.f. of A381751.