A295406 a(n) = n! * Laguerre(n, 2*n, -n).

1, 4, 58, 1422, 49000, 2174360, 118023264, 7574532826, 561071549056, 47111034709260, 4421715905632000, 458741213603157254, 52129735913348001792, 6439324687323193520608, 859089518697047400878080, 123108032319553206480143250, 18858657171509448248927617024

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, A295407, A295408.

Programs

Magma

[Factorial(n)*(&+[Binomial(3*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,2*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[3*n, n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]

PARI

for(n=0,30, print1(n!*sum(k=0, n, binomial(3*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

PARI

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

Formula

a(n) = n!*Sum_{k=0..n} binomial(3*n,n-k)*n^k/k!.

a(n) ~ sqrt(1/2 + 5/(2*sqrt(13))) * ((11 + sqrt(13))/2)^n * exp((sqrt(13)-5)*n/2) * n^n.

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