A289214 a(n) = n! * Laguerre(n,-8).

1, 9, 98, 1238, 17688, 280888, 4894672, 92676144, 1891712384, 41361536384, 963532779264, 23806296960256, 621407739476992, 17078348640463872, 492724294444623872, 14883442560920164352, 469591293625846038528, 15443081743064125505536, 528340631093887891603456

Offset: 0

Author

Alois P. Heinz, Jun 28 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..431
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Column k=8 of A289192.

Cf. A160603, A160604.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(8*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 11 2018

Maple

a:= n-> n! * add(binomial(n, i)*8^i/i!, i=0..n):
seq(a(n), n=0..20);

Mathematica

Table[n!*LaguerreL[n, -8], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)

PARI

my(x = 'x + O('x^30)); Vec(serlaplace(exp(8*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017

PARI

a(n) = n!*pollaguerre(n, 0, -8); \\ Michel Marcus, Feb 05 2021

Python

from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -8))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

Formula

E.g.f.: exp(8*x/(1-x))/(1-x).

a(n) = n! * Sum_{i=0..n} 8^i/i! * binomial(n,i).

a(n) = n! * A160603(n)/A160604(n).

a(n) ~ exp(-4 + 4*sqrt(2*n) - n) * n^(n + 1/4) / 2^(5/4) * (1 + 451/(96*sqrt(2*n))). - Vaclav Kotesovec, Nov 13 2017

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 8^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020