A382794 a(n) = Sum_{k=0..n} Stirling1(n,k) * Stirling2(n,k) * (k!)^2.

1, 1, 3, 2, -418, -14676, -234344, 18565056, 2659703616, 169046742960, -6539356064736, -4061128974843744, -672969012637199040, -19289566159655581440, 27323548725052131528960, 10157639436460221570630144, 1433264952547826545065237504, -520046813680980959472490690560

Offset: 0

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

• sign

Crossrefs

Cf. A000670, A006252, A047792, A048144, A064618, A192554, A192564, A382792.

Programs

Mathematica

Table[Sum[StirlingS1[n, k] StirlingS2[n, k] (k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(1 - (Exp[x] - 1) Log[1 + y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - (exp(x) - 1) * log(1 + y)).