A295418 a(n) = n! * Laguerre(n, n*(n-1), -n).

1, 2, 32, 1422, 124832, 18246850, 4005713952, 1232956594814, 506672220394496, 267992015325604578, 177340024595660672000, 143531889358151618790862, 139482579412432078779322368, 160267575964062522718064075618, 214924620455826226723051817295872

Offset: 0

Author

Vaclav Kotesovec, Nov 22 2017

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..213
• 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, A295409.

Programs

Magma

[Factorial(n)*(&+[Binomial(n^2, n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, May 13 2018

Mathematica

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

PARI

for(n=0,25, print1(n!*sum(k=0,n, binomial(n^2, n-k)*n^k/k!), ", ")) \\ G. C. Greubel, May 13 2018

PARI

a(n) = n!*pollaguerre(n, n*(n-1), -n); \\ Michel Marcus, Feb 05 2021

Formula

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

a(n) ~ exp(1/2) * n^(2*n).

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