OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..366
Wikipedia, Laguerre polynomials
FORMULA
a(n) = (2*n+1)! * Sum_{k=0..n} (-1)^k * binomial(n,k)/(k+n+1)!.
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(2*n+1,k)/(n-k)!.
a(n) = n! * LaguerreL(n, n+1, 1).
a(n) = n! * [x^n] exp(-x/(1 - x))/(1 - x)^(n+2).
a(n) ~ 2^(2*n + 3/2) * n^n / exp(n+1). - Vaclav Kotesovec, May 03 2021
MATHEMATICA
a[n_] := n!*LaguerreL[n, n + 1, 1]; Array[a, 19, 0] (* Amiram Eldar, May 11 2021 *)
PROG
(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k!*binomial(n, k)*binomial(2*n+1, k));
(PARI) a(n) = (2*n+1)!*sum(k=0, n, (-1)^k*binomial(n, k)/(k+n+1)!);
(PARI) a(n) = n!*sum(k=0, n, (-1)^(n-k)*binomial(2*n+1, k)/(n-k)!);
(PARI) a(n) = n!*pollaguerre(n, n+1, 1);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 03 2021
STATUS
approved