A380648 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-4*x)/(1 + x)^4 ).

1, 8, 188, 7816, 475096, 38289504, 3857806144, 467330651456, 66209818738176, 10747317030192640, 1967261819870112256, 400989528160028255232, 90087157573721153554432, 22119056538323287540637696, 5893098619063477612068864000, 1693364632974231188010697990144

Offset

0,2

Links

Table of n, a(n) for n=0..15.

Index entries for reversions of series

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^4 * exp(4 * x * A(x)).

a(n) = 4 * n! * Sum_{k=0..n} (4*n+4)^(k-1) * binomial(4*n+4,n-k)/k!.

a(n) ~ sqrt(58/sqrt(65)-6) * 2^(9*n+6) * exp(((sqrt(65)-9)*n + sqrt(65) - 7)/2) * n^(n-1) / (101*sqrt(65)-773)^(n+1). - Vaclav Kotesovec, Jan 31 2026

Mathematica

nmax=16; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-4*x]/(1 + x)^4, {x, 0, nmax}]], x]Range[0, nmax-1]! (* Stefano Spezia, Feb 06 2025 *)

Prog

(PARI) a(n) = 4*n!*sum(k=0, n, (4*n+4)^(k-1)*binomial(4*n+4, n-k)/k!);

Crossrefs

Cf. A088690, A380646, A380647.

Cf. A370058.

Sequence in context: A163701 A354407 A244438 * A272529 A198282 A177147

Adjacent sequences: A380645 A380646 A380647 * A380649 A380650 A380651

Keyword

nonn

Author

Seiichi Manyama, Feb 06 2025

Status

approved