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A305987 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k).

%I A305987 #10 Feb 16 2025 08:33:54
%S A305987 1,1,2,9,52,355,2976,29897,343988,4423503,63088600,992691205,
%T A305987 17095554444,319404545291,6427307831504,138546745515393,
%U A305987 3185841858310180,77866726065935239,2016161715005701128,55127056896177521981,1587073087715010466556,47982707153606476112067,1519931218769637781731712
%N A305987 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k).
%C A305987 Stirling transform of A007838.
%H A305987 Vaclav Kotesovec, <a href="/A305987/b305987.txt">Table of n, a(n) for n = 0..420</a>
%H A305987 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305987 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A305987 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*(exp(x) - 1)^(j*k)/(k*j^k)).
%F A305987 a(n) = Sum_{k=0..n} Stirling2(n,k)*A007838(k).
%F A305987 a(n) ~ exp(-gamma) * n! / (2 * log(2)^(n+1)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jul 23 2019
%p A305987 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305987       add(combinat[multinomial](n, n-i*j, i$j)/j!*
%p A305987       b(n-i*j, i-1)*(i-1)!^j, j=0..min(1, n/i))))
%p A305987     end:
%p A305987 a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
%p A305987 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 15 2018
%t A305987 nmax = 22; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A305987 nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A305987 b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, (-#)^(1 - k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]
%Y A305987 Cf. A007838, A305547, A305550, A305986.
%K A305987 nonn
%O A305987 0,3
%A A305987 _Ilya Gutkovskiy_, Jun 15 2018