OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..435
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
FORMULA
E.g.f.: exp(5*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 5^i/i! * binomial(n,i).
a(n) ~ exp(-5/2 + 2*sqrt(5*n) - n) * n^(n + 1/4) / (sqrt(2)*5^(1/4)) * (1 + 223/(48*sqrt(5*n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 5^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020
MAPLE
a:= n-> n! * add(binomial(n, i)*5^i/i!, i=0..n):
seq(a(n), n=0..20);
MATHEMATICA
[Table[n!*LaguerreL[n, -5], {n, 0, 50}]] (* G. C. Greubel, May 09 2018 *)
PROG
(PARI) for(n=0, 30, print1(n!*sum(k=0, n, binomial(n, k)*(5^k/k!)), ", ")) \\ G. C. Greubel, May 09 2018
(PARI) a(n) = n!*pollaguerre(n, 0, -5); \\ Michel Marcus, Feb 05 2021
(Magma) [(Factorial(n)*(&+[Binomial(n, k)*(5^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 09 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 28 2017
STATUS
approved