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%I #13 Aug 19 2018 03:52:38
%S 1,1,6,57,748,12565,257526,6232765,173980920,5502613833,194477548330,
%T 7596028355641,324920533473108,15106155118606045,758463525318426942,
%U 40901033617318501845,2357682497456804486896,144670077586483815863569,9414952083720893890165842,647715776085173413399687633
%N Expansion of e.g.f. 1/(1 + LambertW(-x/(1 - x))).
%C Lah transform of A000312.
%H Robert Israel, <a href="/A305276/b305276.txt">Table of n, a(n) for n = 0..369</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = Sum_{k=0..n} binomial(n-1,k-1)*k^k*n!/k!.
%F a(n) ~ n^n * (1 + exp(1))^(n - 1/2) / exp(n - 1/2). - _Vaclav Kotesovec_, Aug 18 2018
%p S:= series(1/(1+LambertW(-x/(1-x))),x,51):
%p seq(coeff(S,x,j)*j!,j=0..50); # _Robert Israel_, Aug 19 2018
%t nmax = 19; CoefficientList[Series[1/(1 + LambertW[-x/(1 - x)]), {x, 0, nmax}], x] Range[0, nmax]!
%t Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^k n!/k!, {k, n}], {n, 19}]]
%Y Cf. A000312, A052871, A060356, A305304.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 18 2018