A346888 Expansion of e.g.f. 1 / (1 - x^2 * exp(x) / 2).

1, 0, 1, 3, 12, 70, 465, 3591, 31948, 319068, 3539385, 43205635, 575312826, 8298867798, 128921967265, 2145837600375, 38097353658120, 718657756980376, 14354000800751313, 302625047150614179, 6716038666999745710, 156498725047355717250, 3820426102008414736761

Offset: 0

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..442

Crossrefs

Column k=2 of A351703.

Cf. A006153, A346889, A346890, A346893.

Cf. A000217, A133189, A308946, A353998.

Programs

Mathematica

nmax = 22; CoefficientList[Series[1/(1 - x^2 Exp[x]/2), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

PARI

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^2*exp(x)/2))) \\ Michel Marcus, Aug 06 2021

PARI

a(n) = n!*sum(k=0, n\2, k^(n-2*k)/(2^k*(n-2*k)!)); \\ Seiichi Manyama, May 13 2022

Formula

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

a(n) ~ n! / ((1 + LambertW(1/sqrt(2))) * 2^(n+1) * LambertW(1/sqrt(2))^n). - Vaclav Kotesovec, Aug 08 2021

a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k)/(2^k * (n-2*k)!). - Seiichi Manyama, May 13 2022