A380647 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x)/(1 + x)^3 ).

1, 6, 105, 3246, 146637, 8780688, 657224901, 59140486800, 6223651526457, 750357182131200, 102014741343847329, 15443915464974191616, 2576937457466957107845, 469914373917914931984384, 92982800086882512621716925, 19843243096453465663599962112, 4543276116844426827394718716401

Offset: 0

Author

Seiichi Manyama, Feb 06 2025

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A088690, A380646, A380648.

Cf. A370055, A377830.

Cf. A377893.

Programs

Mathematica

nmax=17; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-3*x]/(1 + x)^3 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)

PARI

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

Formula

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

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A377893.

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