A380830 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x) ).

1, 4, 47, 978, 29769, 1201728, 60656679, 3681441648, 261337079601, 21256149703680, 1949700750690879, 199146039242552064, 22420399033075845177, 2758645779752490872832, 368321963942753147683575, 53038788218443786432223232, 8194316429830951008255159009, 1352065789150879084276947222528

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Index entries for reversions of series

Formula

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

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

a(n) = A376094(n+1)/(n+1).

a(n) ~ (5 + sqrt(21))^(n+1) * n^(n-1) / (3^(3/4) * 7^(1/4) * 2^(n+1) * exp(((5 - sqrt(21))*n + 3 - sqrt(21))/2)). - Vaclav Kotesovec, Feb 01 2026

Mathematica

nterms=18; CoefficientList[(1/x)*InverseSeries[Series[x*Exp[-3*x]/(1 + x), {x, 0, nterms}], x], x]*Range[0, nterms-1]! (* Stefano Spezia, Nov 11 2025 *)

Prog

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

Crossrefs

Cf. A361182, A380826, A380829.

Cf. A088690, A380828.

Cf. A376094.

Sequence in context: A319833 A006438 A389724 * A251665 A210828 A361560

Adjacent sequences: A380827 A380828 A380829 * A380831 A380832 A380833

Keyword

nonn

Author

Seiichi Manyama, Feb 05 2025

Status

approved