A382830 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * |Stirling1(n,k)| * k!.

1, 1, 8, 102, 1804, 40890, 1131108, 36948240, 1391945616, 59411849040, 2833582748160, 149347596487056, 8620256620495584, 540775669746661440, 36636074309252234880, 2665704585421541790720, 207329122282259073044736, 17165075378189396045777280, 1507206260097615729874083840

Offset: 0

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

• nonn

Crossrefs

Cf. A007840, A052801, A277759, A305919, A354122, A354123.

Programs

Mathematica

Table[Sum[Binomial[n + k - 1, k] Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[1/(1 + Log[1 - x])^n, {x, 0, n}], {n, 0, 18}]

Formula

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

a(n) ~ LambertW(exp(2))^n * n^n / (sqrt(1 + LambertW(exp(2))) * exp(n) * (LambertW(exp(2)) - 1)^(2*n)). - Vaclav Kotesovec, Apr 06 2025