OFFSET
0,2
COMMENTS
In general, for fixed m >= 1, n! * Sum_{k=0..n} binomial(m*n, n-k) * n^k / k! = n! * Laguerre(n, (m-1)*n, -n) ~ sqrt(1/2 + (m + 2)/(2*sqrt(m^2 + 4))) * (2^(m+1) * m^m / ((sqrt(m^2 + 4) - m) * (m - 2 + sqrt(m^2 + 4))^m))^n * exp((sqrt(m^2 + 4) - m)*n/2 - n) * n^n.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
FORMULA
a(n) = n!*Sum_{k=0..n} binomial(5*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 7/(2*sqrt(29))) * (131 - 22*sqrt(29))^n * exp((sqrt(29)-7)*n/2) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(4*n+1). - Ilya Gutkovskiy, Nov 23 2017
MATHEMATICA
Table[n!*LaguerreL[n, 4*n, -n], {n, 0, 15}]
Join[{1}, Table[n!*Sum[Binomial[5*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(5*n, n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018
(PARI) a(n) = n!*pollaguerre(n, 4*n, -n); \\ Michel Marcus, Feb 05 2021
(Magma) [Factorial(n)*(&+[Binomial(5*n, n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Nov 22 2017
STATUS
approved