A372315 Expansion of e.g.f. exp( x - LambertW(-2*x)/2 ).

1, 2, 8, 68, 960, 18832, 471136, 14324480, 512733696, 21119803136, 984029612544, 51169331031040, 2937675286583296, 184560174104465408, 12594824112085327872, 927757127285523243008, 73369903633161123397632, 6200198958236463387836416

Offset: 0

Author

Seiichi Manyama, Apr 27 2024

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A088957, A360193, A372316, A372320, A372321.

Cf. A362694.

Programs

PARI

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

PARI

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

Formula

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

G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k / (1-x)^(k+1).

a(n) ~ 2^(n-1) * n^(n-1) * exp((exp(-1) + 1)/2). - Vaclav Kotesovec, May 04 2024