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%I #16 Mar 26 2019 11:53:44
%S 1,1,5,28,197,1676,16597,186796,2350105,32634928,495207881,8144456684,
%T 144204493765,2733218222944,55188182951917,1182163846918156,
%U 26765995313355953,638508459302742464,16002492517241163793,420279349847440766284,11540406000681962458141,330624627443307824367616
%N Expansion of e.g.f. exp(x*exp(x)/(1 - x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} A007526(k)*binomial(n-1,k-1)*a(n-k).
%F a(n) ~ exp(1/4 - 3*exp(1)/2 + 2*exp(1/2)*sqrt(n) - n) * n^(n - 1/4) / sqrt(2). - _Vaclav Kotesovec_, Aug 25 2018
%p a:=series(exp(x*exp(x)/(1 - x)), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # _Paolo P. Lava_, Mar 26 2019
%t nmax = 21; CoefficientList[Series[Exp[x Exp[x]/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Sum[Floor[Exp[1] k! - 1] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
%o (PARI) x = 'x + O('x^25); Vec(serlaplace(exp(x*exp(x)/(1 - x)))) \\ _Michel Marcus_, Aug 25 2018
%Y Cf. A000248, A000262, A000522, A007526, A318365.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 24 2018
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