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A330354 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)).

%I A330354 #12 Dec 16 2019 07:58:26
%S A330354 1,2,1,21,-122,1752,-21730,309166,-4521032,70344768,-1173530712,
%T A330354 21642745704,-448130571696,10352684535840,-260101132095888,
%U A330354 6921279885508848,-191813249398678272,5502934340821289088,-163695952380982280832,5078687529186002247552
%N A330354 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)).
%H A330354 Vaclav Kotesovec, <a href="/A330354/b330354.txt">Table of n, a(n) for n = 1..400</a>
%F A330354 E.g.f.: -Sum_{k>=1} log(1 - log(1 + x)^k).
%F A330354 E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306042.
%F A330354 exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of the partition numbers (A000041).
%F A330354 a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * sigma(k), where sigma = A000203.
%F A330354 Conjecture: a(n) ~ n! * (-1)^n * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - _Vaclav Kotesovec_, Dec 16 2019
%t A330354 nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t A330354 Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]
%Y A330354 Cf. A000041, A000203, A002743, A008275, A038048, A089064, A306042, A330351, A330352, A330353, A330494.
%K A330354 sign
%O A330354 1,2
%A A330354 _Ilya Gutkovskiy_, Dec 11 2019