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
Keywords
Programs
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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
Comments