A290158 a(n) = n! * [x^n] exp(-n*x)/(1 + LambertW(-x)).

1, 0, 4, -9, 208, -1525, 33516, -463099, 11293248, -231839577, 6517863100, -175791146311, 5723314711632, -189288946716181, 7083626583237036, -275649085963046475, 11724766124450058496, -522717581675749841713, 24981438186138642481404

Offset: 0

Author

Ilya Gutkovskiy, Oct 06 2017

Keywords

• sign

Comments

The n-th term of the n-th inverse binomial transform of A000312.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..385
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000312, A069856, A290840, A318615, A356806.

Programs

Mathematica

Table[n! SeriesCoefficient[Exp[-n x]/(1 + LambertW[-x]), {x, 0, n}], {n, 0, 18}]

PARI

a(n) = (-1)^n*n!*sum(k=0, n\2, n^k*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, May 05 2023

Formula

a(n) ~ (-1)^n * n^n / (1 + LambertW(1)). - Vaclav Kotesovec, Oct 06 2017

From Seiichi Manyama, May 05 2023: (Start)

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

a(n) = (-1)^n * n! * Sum_{k=0..floor(n/2)} n^k * Stirling2(n-k,k)/(n-k)!.

a(n) = [x^n] Sum_{k>=0} (k*x)^k / (1 + n*x)^(k+1).

a(n) = Sum_{k=0..n} (-n)^(n-k) * k^k * binomial(n,k). (End)