A382808 a(n) = Sum_{k=0..n} (|Stirling1(n,k)| * k!)^3.

1, 1, 9, 440, 71344, 25826824, 17321581592, 19304140340736, 33142988156751360, 82906630912116006912, 289508760665893747703808, 1364207202603804952193826816, 8438589244471363680258331914240, 66972265137135031645961782287814656, 668922701586813036491303458870218731520

Offset: 0

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

• nonn

Comments

In general, for m>=1, Sum_{k=0..n} (abs(Stirling1(n,k)) * k!)^m ~ sqrt(2*Pi/m) * n^(m*n + 1/2) / (exp(1) - 1)^(m*n+1). - Vaclav Kotesovec, Apr 05 2025

Crossrefs

Cf. A007840, A242280, A382792, A382807.

Programs

Mathematica

Table[Sum[(Abs[StirlingS1[n, k]] k!)^3, {k, 0, n}], {n, 0, 14}]
Table[(n!)^3 SeriesCoefficient[1/(1 + Log[1 - x] Log[1 - y] Log[1 - z]), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 14}]

Formula

a(n) = (n!)^3 * [(x*y*z)^n] 1 / (1 + log(1 - x) * log(1 - y) * log(1 - z)).

a(n) ~ sqrt(2*Pi/3) * n^(3*n + 1/2) / (exp(1) - 1)^(3*n+1). - Vaclav Kotesovec, Apr 05 2025