%I #11 May 06 2024 05:35:23

%S -1,0,4,36,464,8560,206112,6104896,214376192,8701657344,400748710400,

%T 20642974511104,1175888936749056,73389707156586496,

%U 4980134850525986816,365062349226075463680,28747688571714736160768,2420266280392895064506368

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

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

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

%F a(n) ~ 2^(n-1) * n^(n-1) * exp((exp(-1) - 1)/2). - _Vaclav Kotesovec_, May 06 2024

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

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

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

%K sign

%O 0,3

%A _Seiichi Manyama_, Apr 27 2024