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%I #10 Aug 31 2024 06:19:04
%S 1,0,0,6,0,60,1080,840,80640,982800,5292000,249812640,2854051200,
%T 46711304640,1595483809920,22132648137600,649972279756800,
%U 19151306772998400,377272414943424000,14076577060273728000,407012458114918656000,11429334092933569612800
%N E.g.f. satisfies A(x) = exp(x * (exp(x^2*A(x)) - 1)).
%F a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(n-2*k-1) * Stirling2(k,n-2*k)/k!.
%F a(n) ~ sqrt((s + (2-r)*r^2*s^2) / (1 + r^2*s)) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.61449673663401194313060646272783564740280675129432866295196... and s = 2.0142668139632529702005737408942958028763507472726001354659... are real roots of the system of equations exp((-1 + exp(r^2*s))*r) = s, exp(r^2*s)*s*r^3 = 1. - _Vaclav Kotesovec_, Aug 31 2024
%t Table[n! * Sum[(k+1)^(n-2*k-1) * StirlingS2[k,n-2*k]/k!, {k,0,Floor[n/2]}], {n,0,20}] (* _Vaclav Kotesovec_, Aug 31 2024 *)
%o (PARI) a(n) = n!*sum(k=0, n\2, (k+1)^(n-2*k-1)*stirling(k, n-2*k, 2)/k!);
%Y Cf. A349560, A375830.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Aug 30 2024