A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).

0, 0, 1, 3, 9, 40, 225, 1491, 11578, 102852, 1026945, 11394955, 139091106, 1852061718, 26716291693, 415033647315, 6908006807640, 122645325067576, 2313546734841633, 46209268921868595, 974228913850588750, 21620679147700290210, 503810188866302511501

Offset: 0

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

• nonn

Crossrefs

Cf. A000217, A009444, A133189, A292953, A305990, A346750, A346754, A346755.

Programs

Mathematica

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

Formula

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

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

a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023