A343896 - OEIS

%I #21 May 11 2021 01:54:09

%S 1,2,11,104,1405,24694,534223,13719404,407730041,13760958410,

%T 519827337331,21726980525392,995403499490101,49600090942276094,

%U 2670566242480261175,154500457959360271124,9557826199486960327153,629586464929967678553874,43994787057844036765113691

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

%H Seiichi Manyama, <a href="/A343896/b343896.txt">Table of n, a(n) for n = 0..366</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

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

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

%F a(n) = n! * LaguerreL(n, n+1, 1).

%F a(n) = n! * [x^n] exp(-x/(1 - x))/(1 - x)^(n+2).

%F a(n) ~ 2^(2*n + 3/2) * n^n / exp(n+1). - _Vaclav Kotesovec_, May 03 2021

%t a[n_] := n!*LaguerreL[n, n + 1, 1]; Array[a, 19, 0] (* _Amiram Eldar_, May 11 2021 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k!*binomial(n, k)*binomial(2*n+1, k));

%o (PARI) a(n) = (2*n+1)!*sum(k=0, n, (-1)^k*binomial(n, k)/(k+n+1)!);

%o (PARI) a(n) = n!*sum(k=0, n, (-1)^(n-k)*binomial(2*n+1, k)/(n-k)!);

%o (PARI) a(n) = n!*pollaguerre(n, n+1, 1);

%Y Cf. A006902, A343832, A343890.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 03 2021