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
Keywords
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
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) ~ 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