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%I #16 Mar 10 2024 08:01:17
%S 1,0,2,6,12,140,1470,10122,114296,1874952,25462170,379431470,
%T 7546461252,151797222876,3066316693622,72101615826450,
%U 1843378516587120,47860832586054032,1338908395558366386,40675047500003794902,1282380661224172506620
%N E.g.f. satisfies A(x) = 1 + x^2*exp(x*A(x)).
%F a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k) * binomial(n-2*k+1,k)/( (n-2*k+1)*(n-2*k)! ).
%F From _Vaclav Kotesovec_, Mar 10 2024: (Start)
%F E.g.f.: 1 - LambertW(-exp(x)*x^3)/x.
%F a(n) ~ sqrt(1 + LambertW(exp(-1/3)/3)) * n^(n-1) / (exp(n) * 3^(n + 1/2) * LambertW(exp(-1/3)/3)^(n+1)). (End)
%t nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^3]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Mar 10 2024 *)
%o (PARI) a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(n-2*k+1, k)/((n-2*k+1)*(n-2*k)!));
%Y Cf. A370984, A371018, A371043.
%Y Cf. A161631, A371044.
%Y Cf. A364978.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Mar 09 2024
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