A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

1, 0, 2, 6, 72, 620, 8640, 122346, 2156672, 41367672, 905126400, 21646532270, 570077595648, 16268377195044, 502096929431552, 16629319748711250, 588938142209310720, 22196966267762213744, 887352465220427317248, 37496112562144553167062, 1670071417348195942400000, 78195398849926292810318940

Offset: 0

Author

Ilya Gutkovskiy, Nov 16 2017

Keywords

• nonn

Comments

The n-th term of the n-fold exponential convolution of A000166 with themselves.

Links

• Robert Israel, Table of n, a(n) for n = 0..396
• N. J. A. Sloane, Transforms
• Index entries for sequences related to factorial numbers

Crossrefs

Main diagonal of A295181.

Cf. A000166, A000407, A001813, A087981, A137775, A295183, A383379.

Programs

Maple

S:= series((exp(-x)/(1-x))^n,x,30):
seq(n!*coeff(S,x,n),n=0..29); # Robert Israel, Nov 16 2017

Mathematica

Table[n! SeriesCoefficient[Exp[-n x]/(1 - x)^n, {x, 0, n}], {n, 0, 21}]

Formula

a(n) = A295181(n,n).

a(n) ~ phi^(3*n - 1/2) * n^n / (5^(1/4) * exp(n*(1 + 1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017

a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k-1,k)/(n-k)!. - Seiichi Manyama, Apr 25 2025