A370894 Expansion of e.g.f. (1/x) * Series_Reversion( x*(3 - exp(2*x))/2 ).

1, 1, 6, 64, 1016, 21576, 575680, 18525088, 698625408, 30229271680, 1476535180544, 80371762466304, 4824793854177280, 316685993746640896, 22563822118152880128, 1734427247284290015232, 143072322233503079038976, 12606854482934004152303616

Offset: 0

Author

Seiichi Manyama, Mar 06 2024

Keywords

• nonn

Crossrefs

Cf. A052894, A370934.

Cf. A258922.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(3-exp(2*x))/2)/x))

PARI

a(n) = sum(k=0, n, 2^(n-k)*(n+k)!*stirling(n, k, 2))/(n+1)!;

Formula

a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^(n-k) * (n+k)! * Stirling2(n,k).

a(n) ~ 2^(2*n+1) * LambertW(3*exp(1))^(n+1) * n^(n-1) / (sqrt(1 + LambertW(3*exp(1))) * 3^(n+1) * exp(n) * (LambertW(3*exp(1)) - 1)^(2*n+1)). - Vaclav Kotesovec, Mar 06 2024