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A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

%I A330353 #11 Dec 14 2019 12:04:31
%S A330353 1,4,18,112,810,7144,73458,850672,11069370,161190904,2575237698,
%T A330353 44571447232,836188737930,16970931765064,368985732635538,
%U A330353 8524290269083792,208874053200038490,5428866923032585624,149250273758730282978,4318265042184721248352
%N A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).
%H A330353 Vaclav Kotesovec, <a href="/A330353/b330353.txt">Table of n, a(n) for n = 1..420</a>
%F A330353 E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^k).
%F A330353 E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A167137.
%F A330353 G.f.: Sum_{k>=1} (k - 1)! * sigma(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma = A000203.
%F A330353 exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of the partition numbers (A000041).
%F A330353 a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma(k).
%F A330353 a(n) ~ n! * Pi^2 / (12 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Dec 14 2019
%t A330353 nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t A330353 Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]
%Y A330353 Cf. A000041, A000203, A000629, A002745, A008277, A038048, A167137, A308555, A330351, A330352, A330354.
%K A330353 nonn
%O A330353 1,2
%A A330353 _Ilya Gutkovskiy_, Dec 11 2019