A343896 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * binomial(n,k) * binomial(2*n+1,k).

1, 2, 11, 104, 1405, 24694, 534223, 13719404, 407730041, 13760958410, 519827337331, 21726980525392, 995403499490101, 49600090942276094, 2670566242480261175, 154500457959360271124, 9557826199486960327153, 629586464929967678553874, 43994787057844036765113691

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..366

Wikipedia, Laguerre polynomials

Index entries for sequences related to 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

Cf. A006902, A343832, A343890.

Sequence in context: A099713 A277461 A277470 * A111535 A268294 A364338

Adjacent sequences: A343893 A343894 A343895 * A343897 A343898 A343899

Keyword

nonn

Author

Seiichi Manyama, May 03 2021

Status

approved