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A330450 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)^2).

%I A330450 #9 Dec 16 2019 07:56:26
%S A330450 1,4,7,55,-162,4100,-49030,779914,-11928008,198650880,-3538477560,
%T A330450 70414760136,-1571134087824,38788172175072,-1028732373217200,
%U A330450 28631225505910224,-826097667884640768,24664145505337921920,-765245501125015575168,24841409653689047496576
%N A330450 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)^2).
%H A330450 Vaclav Kotesovec, <a href="/A330450/b330450.txt">Table of n, a(n) for n = 1..400</a>
%F A330450 E.g.f.: -Sum_{k>=1} k * log(1 - log(1 + x)^k).
%F A330450 E.g.f.: log(Product_{k>=1} 1 / (1 - log(1 + x)^k)^k).
%F A330450 exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of A000219.
%F A330450 a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * sigma_2(k), where sigma_2 = A001157.
%F A330450 Conjecture: a(n) ~ n! * (-1)^n * zeta(3) * n * exp(n) / (8 * (exp(1) - 1)^(n+2)). - _Vaclav Kotesovec_, Dec 16 2019
%t A330450 nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t A330450 Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]
%Y A330450 Cf. A000219, A001157, A008275, A318250, A330354, A330449, A330495.
%K A330450 sign
%O A330450 1,2
%A A330450 _Ilya Gutkovskiy_, Dec 15 2019