A307362 Expansion of e.g.f. Sum_{j>=0} (exp(x) - 1)^j / Product_{k=1..j} (1 - k*(exp(x) - 1)).

1, 1, 5, 43, 569, 10651, 265985, 8498323, 336759449, 16158195691, 920970111665, 61390084384003, 4724023128773129, 415070770350493531, 41252331696522595745, 4599993183150111332083, 571422442346267636255609, 78581827113539181495412171, 11896744343184751608550862225

Offset: 0

Author

Ilya Gutkovskiy, Apr 05 2019

Keywords

• nonn

Crossrefs

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

Programs

Mathematica

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]!
nmax = 18; CoefficientList[Series[Sum[j! BellB[j] x^j/Product[(1 - k x), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[StirlingS2[n, k] k! BellB[k], {k, 0, n}], {n, 0, 18}]

Formula

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

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