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%I #12 Mar 10 2024 08:44:36
%S 1,1,2,12,144,1980,31680,630840,15093120,411883920,12607660800,
%T 430740858240,16265744732160,671629503504960,30093198326231040,
%U 1454898560062147200,75503612563771392000,4186035286381024876800,246916968958719605145600
%N E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)^3).
%F a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n+k+1,n-2*k)/( (n+k+1)*k! ).
%F a(n) ~ sqrt((1 + 2*r^2*s^3) / (12*r^2*s + 9*r^4*s^4)) * n^(n-1) / (exp(n) * r^n), where s = 1.766482823850997284176450269002863328615073785089684545740773169... is the root of the equation 3*(s-1)*LambertW(2*s*(s-1)^2) = 2 and r = 1/sqrt(3*s^3*(s-1)) = 0.280882078734447087396397749882018030987007964077248... - _Vaclav Kotesovec_, Mar 10 2024
%t Join[{1}, Table[n! * Sum[(n - 2*k)^k*Binomial[n + k + 1, n - 2*k]/((n + k + 1)*k!), {k, 0, Floor[n/2]}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Mar 10 2024 *)
%o (PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+k+1, n-2*k)/((n+k+1)*k!));
%Y Cf. A358064, A365282, A365283.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 31 2023
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