%I #18 Apr 30 2024 05:45:59

%S 0,1,6,51,684,12965,317298,9500631,336237016,13729172553,635237632350,

%T 32844916975739,1876755685038468,117437155609780461,

%U 7986793018367861194,586578825469711599135,46268265552518066488752,3901008402618593931019409

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

%H Seiichi Manyama, <a href="/A372333/b372333.txt">Table of n, a(n) for n = 0..352</a>

%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=1..n} (2*k)^(k-1) * binomial(n,k).

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

%F a(n) ~ exp(exp(-1)/2) * 2^(n-1) * n^(n-1). - _Vaclav Kotesovec_, Apr 30 2024

%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-exp(x)*lambertw(-2*x)/2)))

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

%Y Cf. A277473, A372334.

%Y Cf. A360548, A372315.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 28 2024