A380081 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + 3*x*exp(x))^(1/3) ).

1, 1, 2, 7, 36, 245, 2086, 21357, 255704, 3507625, 54258570, 934600601, 17743468612, 368146983789, 8288468950958, 201258635444245, 5243025162331056, 145871455305823697, 4316920830720239122, 135408946029576741297, 4487574630295937337500, 156686063319198543135061

Offset: 0

Author

Seiichi Manyama, Jan 11 2025

Keywords

• nonn

Crossrefs

Cf. A161633, A380080.

Cf. A380051.

Programs

PARI

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

Formula

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

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