A351762 Expansion of e.g.f. 1/(1 - 2*x*exp(x)).

1, 2, 12, 102, 1160, 16490, 281292, 5598110, 127326096, 3257961426, 92625793940, 2896747456262, 98827517418456, 3652643136982970, 145385563800940764, 6200097935648462190, 282035994269804870432, 13631368700936950044578, 697586352315912913754916

Offset: 0

Author

Seiichi Manyama, Feb 18 2022

Keywords

• nonn

Crossrefs

Column k=2 of A351761.

Cf. A000629, A351777.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x*exp(x))))

PARI

a(n) = n!*sum(k=0, n, 2^(n-k)*(n-k)^k/k!);

PARI

a(n) = if(n==0, 1, 2*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));

Formula

a(n) = n! * Sum_{k=0..n} 2^(n-k) * (n-k)^k/k!.

a(0) = 1 and a(n) = 2 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.

a(n) ~ n! / ((1 + LambertW(1/2)) * LambertW(1/2)^n). - Vaclav Kotesovec, Feb 19 2022