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E.g.f. satisfies A(x) = exp(x^2/2 + x * A(x)).
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%I #14 Nov 10 2023 07:37:05

%S 1,1,4,22,182,1996,27412,453160,8767516,194438800,4864250096,

%T 135538060384,4163356010728,139784741268160,5093269640966704,

%U 200170986137297536,8440841773833141008,380153135554220691712,18212499110682362677312

%N E.g.f. satisfies A(x) = exp(x^2/2 + x * A(x)).

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

%F E.g.f.: -LambertW(-x * exp(x^2/2)) / x = exp( x^2/2 - LambertW(-x*exp(x^2/2)) ).

%F a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^(n-k-1) / (2^k * k! * (n-2*k)!).

%F a(n) ~ sqrt(1+LambertW(exp(-2))) * n^(n-1) / (exp(n)*LambertW(exp(-2))^((n+1)/2)). - _Vaclav Kotesovec_, Nov 10 2023

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

%Y Cf. A349562, A362748.

%Y Cf. A143740, A362690.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 02 2023