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%I #8 Sep 19 2019 02:21:01
%S 1,0,2,0,26,120,1922,21840,307946,4251240,63165842,1010729280,
%T 18501318266,391496665560,9265945721762,232411950454320,
%U 5972325812958986,156131611764907080,4208451299935189682,119669466221148348960,3658459009408581118106
%N Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - (1 - exp(x))^k).
%C Stirling-Bernoulli transform of partition numbers (A000041).
%F a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n+1,k+1) * k! * A000041(k).
%t nmax = 20; CoefficientList[Series[Exp[x] Product[1/(1 - (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^k StirlingS2[n + 1, k + 1] k! PartitionsP[k], {k, 0, n}], {n, 0, 20}]
%o (PARI) a(n) = sum(k=0, n, (-1)^k*stirling(n+1, k+1, 2)*k!*numbpart(k)); \\ _Michel Marcus_, Sep 19 2019
%Y Cf. A000041, A167137, A306022, A306042.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Sep 18 2019
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