A382033 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

1, 1, 7, 109, 2653, 88261, 3731581, 191571493, 11576241769, 804996352873, 63324553740121, 5559962513556001, 539015912053933645, 57188111522488589293, 6591136171961660099509, 820029701725988751533341, 109537705061927547203868241, 15635869913619342121140932689

Offset: 0

Author

Seiichi Manyama, Mar 12 2025

Keywords

• nonn

Crossrefs

Cf. A161630, A382032, A382034.

Cf. A001764, A377554.

Programs

PARI

a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (k+1)^(n-k-1)*binomial(3*n, k)/(n-k-1)!));

Formula

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(3*n,k)/(n-k-1)! for n > 0.

Let F(x) be the e.g.f. of A377554. F(x) = log(A(x))/x = B(x*A(x))^3.

E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^3 ) ).