A293146 a(n) = n! * [x^n] exp(x/(1 - n*x)).

1, 1, 5, 73, 2161, 108101, 8201701, 878797165, 126422091713, 23514740267401, 5492576235204901, 1574136880033408241, 543143967119720304625, 222106209904092987888013, 106221716052645457812866501, 58741017143127754662557082901, 37194600833984874761008613195521

Offset: 0

Author

Ilya Gutkovskiy, Oct 01 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..232

Crossrefs

Cf. A000262, A025168, A293145.

Programs

Maple

S:=series(exp(x/(1-n*x)),x,31):
seq(coeff(S,x,n)*n!,n=0..30); # Robert Israel, Oct 01 2017

Mathematica

Table[n! SeriesCoefficient[Exp[x/(1 - n x)], {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[Product[Exp[n^k x^(k + 1)], {k, 0, n}], {x, 0, n}], {n, 1, 16}]]
Join[{1}, Table[Sum[n^(n - k) n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 16}]]
Join[{1}, Table[n^n (n - 1)! Hypergeometric1F1[1 - n, 2, -1/n], {n, 1, 16}]]

PARI

{a(n) = if(n==0, 1, n!*sum(k=1, n, n^(n-k)*binomial(n-1, k-1)/k!))} \\ Seiichi Manyama, Feb 03 2021

Formula

a(n) ~ BesselI(1, 2) * sqrt(2*Pi) * n^(2*n-1/2) / exp(n). - Vaclav Kotesovec, Oct 01 2017

a(n) = n! * Sum_{k=1..n} n^(n-k) * binomial(n-1,k-1)/k! for n > 0. - Seiichi Manyama, Feb 03 2021