A381753 Expansion of exp( Sum_{k>=1} binomial(5*k-1,2*k-1) * x^k/k ).

1, 4, 50, 846, 16495, 349240, 7803823, 181135830, 4324897697, 105543188190, 2620784850325, 66005699547352, 1682046970846570, 43291586055360034, 1123707191010320955, 29382536610737191930, 773229801368332554273, 20463493681189771623960

Offset: 0

Author

Seiichi Manyama, Mar 06 2025

Keywords

• nonn
• easy

Crossrefs

Cf. A006013, A079489, A182960, A381751, A381752, A381757, A381758.

Cf. A060941.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(5*k-1, 2*k-1)*x^k/k)))

PARI

a(n) = 2*sum(k=0, n, binomial(5*n+2*k+2, k)*binomial(5*n+2, n-k)/(5*n+2*k+2));

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(5*k-1,2*k-1) * a(n-k).

G.f.: B(x)^2, where B(x) is the g.f. of A060941.

a(n) = 2 * Sum_{k=0..n} binomial(5*n+2*k+2,k) * binomial(5*n+2,n-k)/(5*n+2*k+2).