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E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * x.
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%I #28 Aug 28 2022 04:24:48

%S 1,0,2,3,40,245,2976,35287,524560,8790777,165530800,3493679651,

%T 80812685064,2049413147509,56294089065592,1668771901644135,

%U 53057068616526496,1801519375618579313,65063987978980974048,2490449984485154892235,100716775979173952155480

%N E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * x.

%F a(n) ~ sqrt(s*(1 - r^2*s/(1 + r*s))) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.4599551063707173872728335298048828687860291021728... is the root of the equation r - LambertW(1/r) - 2*log(r) = 1/LambertW(1/r) and s = LambertW(1/r)/r = 1.938208283387405345404104769972407921289092368509... - _Vaclav Kotesovec_, Nov 22 2021

%F a(n) = n! * Sum_{k=0..floor(n/2)} (n-k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. - _Seiichi Manyama_, Aug 27 2022

%p a:= n-> n!*coeff(series(RootOf(A=exp(x*exp(x*A)-x), A), x, n+1), x, n):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Nov 22 2021

%t nmax = 20; A[_] = 0; Do[A[x_] = Exp[(E^(x*A[x]) - 1)*x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 22 2021 *)

%o (PARI) my(A=1,n=22); for(i=1, n, A=exp((exp(x*A)-1)*(x+x*O(x^n)))); Vec(serlaplace(A))

%o (PARI) a(n) = n!*sum(k=0, n\2, (n-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ _Seiichi Manyama_, Aug 27 2022

%Y Cf. A052506, A349557.

%Y Cf. A356785, A356788, A356789.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 22 2021