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A305988 Expansion of e.g.f. 1/(1 + log(2 - exp(x))).

%I A305988 #13 Feb 16 2025 08:33:54
%S A305988 1,1,4,24,194,1970,24062,343294,5601122,102847794,2098766582,
%T A305988 47117285270,1154031484586,30622256174458,875092190716382,
%U A305988 26794239236959806,875110094707912562,30367988674208286914,1115822099409002188358,43276913813553367194598,1766830322476935945014330
%N A305988 Expansion of e.g.f. 1/(1 + log(2 - exp(x))).
%C A305988 Stirling transform of A007840.
%H A305988 Robert Israel, <a href="/A305988/b305988.txt">Table of n, a(n) for n = 0..402</a>
%H A305988 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305988 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A305988 a(n) = Sum_{k=0..n} Stirling2(n,k)*A007840(k).
%F A305988 a(n) ~ n! / ((2*exp(1) - 1) * (log(2 - exp(-1)))^(n+1)). - _Vaclav Kotesovec_, Jul 01 2018
%e A305988 1/(1 + log(2 - exp(x))) = 1 + x + 4*x^2/2! + 24*x^3/3! + 194*x^4/4! + 1970*x^5/5! + 24062*x^6/6! + ...
%p A305988 b:= proc(n) b(n):= n!*`if`(n=0, 1, add(b(k)/(k!*(n-k)), k=0..n-1)) end:
%p A305988 a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
%p A305988 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 15 2018
%t A305988 nmax = 20; CoefficientList[Series[1/(1 + Log[2 - Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
%t A305988 Table[Sum[Sum[StirlingS2[n, k] Abs[StirlingS1[k, j]] j!, {j, 0, k}], {k, 0, n}], {n, 0, 20}]
%Y A305988 Cf. A000629, A000670, A007840, A050351, A083355, A305323.
%K A305988 nonn
%O A305988 0,3
%A A305988 _Ilya Gutkovskiy_, Jun 15 2018