A295409 a(n) = n! * Laguerre(n, n^2, -n).

1, 3, 58, 2859, 267576, 40818095, 9235507968, 2906955312471, 1215257338052992, 651548571287972859, 435901423022852332800, 356000439852418418920643, 348583395952381998326141952, 403108990190536860168604229031, 543577365164816368801494214352896

Offset: 0

Author

Vaclav Kotesovec, Nov 22 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..214
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A277373, A295385, A295406, A295407, A295408, A295418.

Programs

Magma

[Factorial(n)*(&+[Binomial(n*(n+1), n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 11 2018

Maple

seq(n!*orthopoly[L](n,n^2,-n),n=0..30); # Robert Israel, Nov 22 2017

Mathematica

Table[n!*LaguerreL[n,n^2,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[n*(n+1),n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

PARI

for(n=0,30, print1(n!*sum(k=0,30, binomial(n*(n+1), n-k)*n^k/k!), ", ")) \\ G. C. Greubel, May 11 2018

PARI

a(n) = n!*pollaguerre(n, n^2, -n); \\ Michel Marcus, Feb 05 2021

Formula

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

a(n) ~ exp(3/2) * n^(2*n).

a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Nov 23 2017