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%I #20 Apr 15 2023 15:16:48
%S 1,1,5,19,241,1601,32581,308995,8655809,106673761,3805452901,
%T 57704760851,2500580809585,45018720191329,2295683481085541,
%U 47848514992963651,2806491306922172161,66464103165835330625,4407449313521981148229,116893033842508769526931
%N a(n) = n! * Sum_{k=0..floor(n/2)} n^k * binomial(n-k,k)/(n-k)!.
%H Seiichi Manyama, <a href="/A362281/b362281.txt">Table of n, a(n) for n = 0..395</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = n! * [x^n] exp(x + n*x^2).
%F E.g.f.: exp( sqrt( -LambertW(-2*x^2)/2 ) ) / (1 + LambertW(-2*x^2)).
%F a(n) ~ (1 + (-1)^n/exp(sqrt(2))) * 2^((n-1)/2) * n^n / exp(n/2 - 1/sqrt(2)). - _Vaclav Kotesovec_, Apr 15 2023
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sqrt(-lambertw(-2*x^2)/2))/(1+lambertw(-2*x^2))))
%Y Cf. A277614, A362282.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Apr 14 2023