A385750 a(n) = Sum_{k=0..n} Stirling2(n,k) * (n!/k!)^2.

1, 1, 5, 64, 1681, 78651, 5891041, 653545390, 101785047169, 21431911982437, 5927319770834701, 2101574777340578156, 935265924020629176625, 512945332353359967175999, 341342159773993944429746793, 272012935493149854994361194426, 256689188247205271953044107166721, 284051735653584424779666013789038985

Offset: 0

Author

Ilya Gutkovskiy, Jul 08 2025

Keywords

• nonn

Crossrefs

Cf. A000110, A064618, A119392, A119400, A385751, A385752.

Programs

Mathematica

Table[Sum[StirlingS2[n, k] (n!/k!)^2, {k, 0, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Sum[(Exp[x] - 1)^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3

Formula

Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{k>=0} (x^k / k!^2) * Product_{j=1..k} 1 / (1 - j*x).

Sum_{n>=0} a(n) * x^n / n!^3 = Sum_{k>=0} (exp(x) - 1)^k / k!^3.