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A330449 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)^2).

%I A330449 #11 Dec 16 2019 07:56:54
%S A330449 1,6,36,282,2460,25506,299796,3921882,56977740,913248786,15917884356,
%T A330449 299358495882,6066180049020,131932872768066,3057940695635316,
%U A330449 75151035318996282,1954299203147952300,53684552455571903346,1553161560008013680676,47162101103528811791082
%N A330449 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)^2).
%H A330449 Vaclav Kotesovec, <a href="/A330449/b330449.txt">Table of n, a(n) for n = 1..400</a>
%F A330449 E.g.f.: -Sum_{k>=1} k * log(1 - (exp(x) - 1)^k).
%F A330449 E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306046.
%F A330449 G.f.: Sum_{k>=1} (k - 1)! * sigma_2(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma_2 = A001157.
%F A330449 exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000219.
%F A330449 a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma_2(k).
%F A330449 a(n) ~ n! * zeta(3) * n / (4 * (log(2))^(n+2)). - _Vaclav Kotesovec_, Dec 15 2019
%t A330449 nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t A330449 Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]
%Y A330449 Cf. A000219, A001157, A008277, A064602, A306046, A318250, A330353, A330450.
%K A330449 nonn
%O A330449 1,2
%A A330449 _Ilya Gutkovskiy_, Dec 15 2019