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Expansion of e.g.f. Sum_{j>=0} (exp(x) - 1)^j / Product_{k=1..j} (1 - k*(exp(x) - 1)).
1

%I #6 Apr 05 2019 17:46:49

%S 1,1,5,43,569,10651,265985,8498323,336759449,16158195691,920970111665,

%T 61390084384003,4724023128773129,415070770350493531,

%U 41252331696522595745,4599993183150111332083,571422442346267636255609,78581827113539181495412171,11896744343184751608550862225

%N Expansion of e.g.f. Sum_{j>=0} (exp(x) - 1)^j / Product_{k=1..j} (1 - k*(exp(x) - 1)).

%F G.f.: Sum_{j>=0} j!*Bell(j)*x^j / Product_{k=1..j} (1 - k*x).

%F a(n) = Sum_{k=0..n} Stirling2(n,k)*k!*Bell(k).

%t nmax = 18; CoefficientList[Series[Sum[(Exp[x] - 1)^j/Product[(1 - k (Exp[x] - 1)), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 18; CoefficientList[Series[Sum[j! BellB[j] x^j/Product[(1 - k x), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

%t Table[Sum[StirlingS2[n, k] k! BellB[k], {k, 0, n}], {n, 0, 18}]

%Y Cf. A000110, A000258, A000670, A137341, A307363.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 05 2019