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

0,2

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

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

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}]]

Prog

(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
(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

Crossrefs

Cf. A277373, A295385, A295406, A295408.

Sequence in context: A348347 A222903 A024069 * A152283 A205344 A270408

Adjacent sequences: A295404 A295405 A295406 * A295408 A295409 A295410

Keyword

nonn

Author

Vaclav Kotesovec, Nov 22 2017

Status

approved