A318615 a(n) = n! * [x^n] 1/(1 - x)^(n*x).

1, 0, 4, 9, 224, 1650, 38664, 540960, 13930496, 291769128, 8598924000, 237964577400, 8082061452288, 275311724996880, 10714824398213376, 430458433091505000, 19007133744632954880, 876046954673290438080, 43416883192646088235008, 2252711496770428822876800, 124040138653975179571200000

Offset: 0

Author

Ilya Gutkovskiy, Aug 30 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A007113, A008275, A053489, A053490, A191415, A318616.

Programs

Mathematica

Table[n! SeriesCoefficient[1/(1 - x)^(n x), {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[(-1)^n n! Sum[n^(n - k) StirlingS1[k, n - k]/k!, {k, n}], {n, 20}]]

Formula

a(n) = n! * [x^n] exp(n*x*Sum_{k>=1} x^k/k).

a(n) = (-1)^n*n! * Sum_{k=0..n} n^(n-k)*Stirling1(k,n-k)/k!.

a(n) ~ n^n / (sqrt(1 - (1-s)*(2-s)*s) * exp(n) * s^n * (1-s)^(s*n - 1)), where s = 0.530402312512063468084914246777198746... is the root of the equation (1-s)*(2 + s + s*log(1-s)) = 1. - Vaclav Kotesovec, Aug 30 2018