A330649 E.g.f.: Product_{k>=1} 1 / (1 - x^k/(k!*(1 - x)^k)).

1, 1, 5, 34, 299, 3226, 41202, 607545, 10153831, 189628750, 3913009178, 88406043991, 2170372901534, 57531498837515, 1637713270797411, 49830222530823615, 1613950394999111903, 55444724259894089718, 2013760368429942861810, 77105255895256112519259

Offset: 0

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

• nonn

Crossrefs

Cf. A005651, A084358, A218482, A320566, A332024.

Programs

Mathematica

nmax = 19; CoefficientList[Series[Product[1/(1 - x^k/(k! (1 - x)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Total[Apply[Multinomial, IntegerPartitions[k], {1}]] n!/k!, {k, 0, n}], {n, 0, 19}]

PARI

seq(n)={Vec(serlaplace(prod(k=1, n, 1 / (1 - x^k/(k!*(1 - x)^k)) + O(x*x^n))))} \\ Andrew Howroyd, Feb 13 2020

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * A005651(k) * n! / k!.

a(n) ~ c * 2^(n-1) * n!, where c = A247551 = 2.52947747207915264818... - Vaclav Kotesovec, Feb 16 2020