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a(n) = Sum_{k=0..n} (2*k)^k * Stirling2(n,k).
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%I #16 Feb 06 2022 05:23:48

%S 1,2,18,266,5506,146602,4772162,183618794,8152995138,410307648938,

%T 23079780216386,1434953808618090,97716253164212034,

%U 7233006174407149866,578233606405444793410,49651123488091636885994,4557474786380802233761090,445324385454834015896585386

%N a(n) = Sum_{k=0..n} (2*k)^k * Stirling2(n,k).

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

%F E.g.f.: 1/(1 + LambertW( 2 * (1 - exp(x)) )), where LambertW() is the Lambert W-function.

%F a(n) ~ n^n / (sqrt(1 + 2*exp(1)) * (log(exp(1) + 1/2) - 1)^(n + 1/2) * exp(n)). - _Vaclav Kotesovec_, Feb 06 2022

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

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(2*(1-exp(x))))))

%Y Cf. A282190, A308490, A351274, A351277.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 05 2022