A372320 Expansion of e.g.f. -exp( x + LambertW(-2*x)/2 ).

-1, 0, 4, 36, 464, 8560, 206112, 6104896, 214376192, 8701657344, 400748710400, 20642974511104, 1175888936749056, 73389707156586496, 4980134850525986816, 365062349226075463680, 28747688571714736160768, 2420266280392895064506368

Offset: 0

Author

Seiichi Manyama, Apr 27 2024

Keywords

• sign

Links

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

Crossrefs

Cf. A088957, A360193, A372315, A372316, A372321.

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 06 2024