A352074 a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).

1, 1, 4, 42, 904, 34070, 2019888, 174588120, 20804747136, 3276218158560, 659664288364800, 165425062846302336, 50574549124825998336, 18520126461205806360144, 8003819275469728355033088, 4031020344281171589447408000, 2340375822778055527109749211136

Offset: 0

Author

Ilya Gutkovskiy, Mar 02 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..233

Crossrefs

Cf. A007840, A092985, A094420, A227917, A317171, A317172, A331690, A352069, A352071.

Cf. A081048.

Programs

Mathematica

Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! (-n)^(n - k), {k, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[1/(1 + Log[1 - n x]/n), {x, 0, n}], {n, 1, 16}]]

PARI

a(n) = sum(k=0, n, stirling(n, k, 1)*k!*(-n)^(n-k)); \\ Michel Marcus, Mar 02 2022

Formula

a(n) = n! * [x^n] 1 / (1 + log(1 - n*x) / n) for n > 0.

a(n) ~ n! * n^(n-2) * (1 + 2*log(n)/n). - Vaclav Kotesovec, Mar 03 2022