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%I #21 Nov 27 2021 05:20:36
%S 1,1,-2,9,-68,740,-10554,185906,-3891320,94259952,-2592071760,
%T 79748398752,-2713685928744,101184283477680,-4102325527316184,
%U 179674073609647080,-8454031849605513024,425281651659459346944,-22777115050468598701248
%N E.g.f. satisfies: A(x)^A(x) = 1 + x.
%H G. C. Greubel, <a href="/A120980/b120980.txt">Table of n, a(n) for n = 0..375</a>
%F E.g.f.: A(x) = log(1+x)/LambertW(log(1+x)).
%F log(A(x)) = LambertW(log(1+x)).
%F E.g.f.: A(x) = 1/G(-x) where G(x) = g.f. of A052813.
%F E.g.f. of A052807 = -log(A(-x)) = -log(1-x)/A(-x).
%F a(n) = Sum_{k=0..n} (-1)^(k+1)*Stirling1(n,k)*(k-1)^(k-1). - _Vladeta Jovovic_, Jul 22 2006
%F |a(n)| ~ exp((exp(-1)-1)*n+3/2) * n^(n-1) / (exp(exp(-1))-1)^(n-1/2). - _Vaclav Kotesovec_, Jul 09 2013
%t CoefficientList[Series[Log[1+x]/LambertW[Log[1+x]], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jul 09 2013 *)
%t Table[StirlingS1[n, 0] + StirlingS1[n, 1] + Sum[(-1)^(k + 1)*StirlingS1[n, k]*(k - 1)^(k - 1), {k, 2, n}], {n,0,50}] (* _G. C. Greubel_, Jun 21 2017 *)
%t CoefficientList[Series[Exp[LambertW[Log[1+x]]], {x, 0, 25}], x]* Range[0, 25]! (* _G. C. Greubel_, Jun 22 2017 *)
%o (PARI) {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0); A[ #A]=-Vec(Ser(A)^Ser(A))[ #A]);n!*A[n+1]}
%o (PARI) x='x+O('x^50); Vec(serlaplace(exp(lambertw(log(1+x))))) \\ _G. C. Greubel_, Jun 22 2017
%Y Cf. A008275, A052813, A052807, A349561, A349583, A349585.
%K sign
%O 0,3
%A _Paul D. Hanna_, Jul 20 2006
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