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%I #7 Apr 22 2019 18:06:24
%S 1,1,5,35,323,3679,49819,781465,13923545,277563617,6118251461,
%T 147715469131,3875706370315,109781717161375,3338229675519803,
%U 108443658227589329,3747688533281296049,137273241169036231105,5311844045472206624005,216505267421266611639667,9270689769095765333645651
%N Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^(1/k).
%F E.g.f.: exp(Sum_{k>=1} d(k)*x^k/(k*(1 - x)^k)), where d(k) is the number of divisors of k (A000005).
%F a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A028342(k)*n!/k!.
%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 323*x^4/4! + 3679*x^5/5! + 49819*x^6/6! + 781465*x^7/7! + 13923545*x^8/8! + ...
%e log(A(x)) = x + 4*x^2/2 + 11*x^3/3 + 27*x^4/4 + 62*x^5/5 + 137*x^6/6 + 296*x^7/7 + 630*x^8/8 + 1326*x^9/9 + ... + A160399(k)*x^k/k + ...
%t nmax = 20; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%Y Cf. A000005, A028342, A103446, A160399, A218482, A307680, A320563.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 21 2019
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