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%I #34 Oct 27 2023 21:01:02
%S 1,1,3,14,96,849,9362,123101,1888016,33066768,651883152,14286514186,
%T 344690210928,9079702374300,259327537407416,7983107543564724,
%U 263518937698466304,9285770278110061664,347916420499685643072,13812127364516107258944,579183295530010157485824
%N Expansion of e.g.f.: A(x) = -(1 + LambertW(-log(1+x))/log(1+x))/x.
%C A033917 gives the coefficients of iterated exponential function defined by y(x) = x^y(x) expanded about x=1.
%H Alois P. Heinz, <a href="/A136461/b136461.txt">Table of n, a(n) for n = 0..400</a> (first 71 terms from Vincenzo Librandi)
%F a(n) = A033917(n+1)/(n+1).
%F E.g.f.: A(x) = (1/x)*Sum_{i>=1} (i+1)^(i-1) * log(1+x)^i/i!.
%F a(n) ~ n^(n-1) / ( exp(n-3/2+exp(-1)/2) * (exp(exp(-1))-1)^(n+1/2) ). - _Vaclav Kotesovec_, Nov 27 2012
%p a:= n-> add(Stirling1(n+1, k)*(k+1)^(k-1), k=0..n+1)/(n+1):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 21 2016
%t CoefficientList[Series[-(1+LambertW[-Log[1+x]]/Log[1+x])/x, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 27 2012 *)
%o (PARI) {a(n)=n!*polcoeff(sum(i=0,n+1,(i+1)^(i-1)*log(1+x +O(x^(n+2) ))^i/i!), n+1)}
%o (PARI) x='x+O('x^30); Vec(serlaplace(-(1+lambertw(-log(1+x))/log(1+x))/x )) \\ _G. C. Greubel_, Feb 19 2018
%Y Cf. A033917.
%Y Row sums of A295027 (shifted).
%Y Main diagonal of A295028 (shifted).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 31 2007
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