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A305547 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k!).

%I A305547 #75 Feb 16 2025 08:33:54
%S A305547 1,1,2,8,37,182,1039,7149,56382,479220,4280247,40406984,410453366,
%T A305547 4539623168,54431372233,695801259947,9312538336475,128985882874288,
%U A305547 1842668013046405,27238267120063415,419396473955088310,6769168354222927254,114837651830425810381,2042782103293394499566
%N A305547 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k!).
%C A305547 Stirling transform of A007837.
%H A305547 Seiichi Manyama, <a href="/A305547/b305547.txt">Table of n, a(n) for n = 0..500</a>
%H A305547 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305547 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A305547 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*(exp(x) - 1)^(j*k)/((j!)^k*k)).
%F A305547 a(n) = Sum_{k=0..n} Stirling2(n,k)*A007837(k).
%p A305547 b:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
%p A305547        d=numtheory[divisors](k))*(n-1)!/(n-k)!*b(n-k), k=1..n))
%p A305547     end:
%p A305547 a:= n-> add(Stirling2(n, k)*b(k), k=0..n):
%p A305547 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 15 2018
%t A305547 nmax = 23; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A305547 nmax = 23; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(j k)/((j!)^k k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A305547 b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, -# (-#!)^(-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, 23}]
%Y A305547 Cf. A007837, A140585, A305987.
%K A305547 nonn
%O A305547 0,3
%A A305547 _Ilya Gutkovskiy_, Jun 15 2018