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%I #18 Dec 16 2019 11:40:57
%S 1,3,12,70,492,4298,43894,514666,6830888,101473632,1664125944,
%T 29858266392,582481147440,12281821373040,278257595964576,
%U 6739505703156192,173785740554811264,4754455742416944000,137571331202872821504,4197696814883284962048
%N a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * tau(k), where tau = A000005.
%H Vaclav Kotesovec, <a href="/A330493/b330493.txt">Table of n, a(n) for n = 1..400</a>
%F E.g.f.: -Sum_{k>=1} log(1 - log(1/(1 - x))^k) / k.
%F a(n) ~ n! * (log(n) + 2*gamma - log(exp(1) - 1)) / (n * (1 - exp(-1))^n), where gamma is the Euler-Mascheroni constant A001620.
%t Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]
%t nmax = 20; Rest[CoefficientList[Series[-Sum[Log[1 - Log[1/(1 - x)]^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]
%o (PARI) a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*numdiv(k)); \\ _Michel Marcus_, Dec 16 2019
%Y Cf. A330351, A330352, A330494, A330495.
%K nonn
%O 1,2
%A _Vaclav Kotesovec_, Dec 16 2019
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