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A330351 Expansion of e.g.f. -Sum_{k>=1} log(1 - (exp(x) - 1)^k) / k.

%I A330351 #11 Dec 15 2019 08:39:05
%S A330351 1,3,11,57,359,2793,25871,273297,3268199,44132313,659178431,
%T A330351 10710083937,189256343639,3636935896233,75228664345391,
%U A330351 1657133255788977,38770903634692679,964609458391250553,25470259163197390751,709595190213796188417
%N A330351 Expansion of e.g.f. -Sum_{k>=1} log(1 - (exp(x) - 1)^k) / k.
%H A330351 Vaclav Kotesovec, <a href="/A330351/b330351.txt">Table of n, a(n) for n = 1..420</a>
%H A330351 Vaclav Kotesovec, <a href="/A330351/a330351.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A330351 E.g.f.: Sum_{i>=1} Sum_{j>=1} (exp(x) - 1)^(i*j) / (i*j).
%F A330351 E.g.f.: log(Product_{k>=1} 1 / (1 - (exp(x) - 1)^k)^(1/k)).
%F A330351 G.f.: Sum_{k>=1} (k - 1)! * tau(k) * x^k / Product_{j=1..k} (1 - j*x), where tau = A000005.
%F A330351 a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * tau(k).
%F A330351 a(n) ~ n! * (log(n) + 2*gamma - log(2) - log(log(2))) / (n * (log(2))^n), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Dec 14 2019
%t A330351 nmax = 20; CoefficientList[Series[-Sum[Log[1 - (Exp[x] - 1)^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t A330351 Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]
%Y A330351 Cf. A000005, A000629, A002746, A008277, A028342, A308554, A318249, A330352, A330353, A330354, A330445.
%K A330351 nonn
%O A330351 1,2
%A A330351 _Ilya Gutkovskiy_, Dec 11 2019