A382030 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x)^2)), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

1, 1, 3, 37, 817, 25741, 1053211, 52957297, 3157457185, 217695187801, 17036331544531, 1491702434847901, 144479729938558609, 15335923797225215653, 1770255543485671432555, 220776904683577075549801, 29582947262972619472787521, 4238424613351537181204589745, 646565304924896452410832170787

Offset: 0

Author

Seiichi Manyama, Mar 12 2025

Keywords

• nonn

Crossrefs

Cf. A212722, A382029, A382031.

Cf. A001764, A382043.

Programs

PARI

a(n) = if(n==0, 1, n!*sum(k=0, n-1, (2*k+1)^(n-k-1)*binomial(n+2*k, k)/((n+2*k)*(n-k-1)!)));

Formula

Let F(x) be the e.g.f. of A382043. F(x) = log(A(x))/x = B(x*A(x)^2).

a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+2*k,k)/((n+2*k) * (n-k-1)!) for n > 0.