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%I #41 Apr 08 2019 03:08:16
%S 0,1,4,33,424,7445,166176,4505053,143787904,5282091081,219531404800,
%T 10184792907641,521761503753216,29254578504622237,1781920872844693504,
%U 117169936148978011125,8272258025961978167296
%N E.g.f. A(x) is inverse to F(x) = x*exp(-x)/(1+x).
%H Robert Israel, <a href="/A052885/b052885.txt">Table of n, a(n) for n = 0..357</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=858">Encyclopedia of Combinatorial Structures 858</a>
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.
%H I. Mezo, A. Baricz, <a href="http://arxiv.org/abs/1408.3999">On the generalization of the Lambert W function with applications in theoretical physics</a>, arXiv preprint arXiv:1408.3999 [math.CA], 2014-2015.
%F E.g.f.: RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z).
%F E.g.f. A(x) = sum(n>0, a(n)*x^n/n!) is inverse to F(x)=x*exp(-x)/(1+x), a(n)=(n-1)!*sum_{i=0..n-1} (n^(n-i-1)*binomial(n,i))/(n-i-1)!, n>0. - _Vladimir Kruchinin_, Jan 31 2012
%F a(n) ~ 5^(-1/4) * ((3+sqrt(5))/2)^n * exp((sqrt(5)-3)*n/2) * n^(n-1). - _Vaclav Kotesovec_, Jan 23 2014
%p spec := [S,{B=Prod(Z,C),C=Set(S),S=Sequence(B,1<= card)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t CoefficientList[InverseSeries[Series[x/(E^x*(1+x)),{x,0,20}],x],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 23 2014 *)
%o (Maxima) a(n):=((n-1)!*sum((n^(n-i-1)*binomial(n,i))/(n-i-1)!,i,0,n-1)); /* _Vladimir Kruchinin_, Jan 31 2012 */
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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