A295407 a(n) = n! * Laguerre(n, 3*n, -n).

1, 5, 92, 2859, 124832, 7018105, 482598720, 39236322839, 3681751480832, 391611920476653, 46560370087846400, 6119025385880816035, 880818377346674454528, 137824220501484017301281, 23291983597732334528110592, 4228010378355969165140319375

Offset: 0

Author

Vaclav Kotesovec, Nov 22 2017

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..300
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A277373, A295385, A295406, A295408.

Programs

Magma

[Factorial(n)*(&+[Binomial(4*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Mathematica

Table[n!*LaguerreL[n,3*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[4*n, n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]

PARI

for(n=0,30, print1(n!*sum(k=0, n, binomial(4*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

PARI

a(n) = n!*pollaguerre(n, 3*n, -n); \\ Michel Marcus, Feb 05 2021

Formula

a(n) = n!*Sum_{k=0..n} binomial(4*n,n-k)*n^k/k!.

a(n) ~ sqrt(1/2 + 3/(2*sqrt(5))) * (8*(sqrt(5)-1))^n * exp((sqrt(5)-3)*n) * n^n.

a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(3*n+1). - Ilya Gutkovskiy, Nov 23 2017