A382792 a(n) = Sum_{k=0..n} (Stirling1(n,k) * k!)^2.

1, 1, 5, 76, 2392, 126676, 10057204, 1114096320, 163918005696, 30894047577216, 7254176241285504, 2075722128162164736, 710883208780304954112, 287061726161439955116288, 134961239570613490548986112, 73079781978184515947237031936, 45150931601954398539342470578176

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

Main diagonal of A379821.

Cf. A006252, A007840, A047796, A048144, A382794.

Programs

Mathematica

Table[Sum[(StirlingS1[n, k] k!)^2, {k, 0, n}], {n, 0, 16}]
Table[(n!)^2 SeriesCoefficient[1/(1 - Log[1 + x] Log[1 + y]), {x, 0, n}, {y, 0, n}], {n, 0, 16}]

PARI

a(n) = sum(k=0, n, (k!*stirling(n, k, 1))^2); \\ Seiichi Manyama, Apr 05 2025

Formula

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

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

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