A351763 Expansion of e.g.f. 1/(1 - 3*x*exp(x)).

1, 3, 24, 279, 4332, 84075, 1958058, 53202387, 1652070696, 57713665779, 2240196853710, 95650311987483, 4455281606078988, 224815388384744859, 12216916158370619010, 711312392929267383075, 44176151714082889756368, 2915038701200389804440675

Offset: 0

Author

Seiichi Manyama, Feb 18 2022

Keywords

• nonn

Crossrefs

Column k=3 of A351761.

Cf. A351778.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*exp(x))))

PARI

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

PARI

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

Formula

E.g.f.: 1/(1 - 3*x*exp(x)).

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

a(0) = 1 and a(n) = 3 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.

a(n) ~ n! / ((1 + LambertW(1/3)) * LambertW(1/3)^n). - Vaclav Kotesovec, Feb 19 2022