A289211 a(n) = n! * Laguerre(n,-5).

1, 6, 47, 446, 4929, 61870, 866695, 13373190, 224995745, 4094022230, 80031878175, 1671426609550, 37116087808225, 872797202471550, 21656891639499575, 565266064058561750, 15476777687220818625, 443409439715399299750, 13263588837009155407375

Offset: 0

Author

Alois P. Heinz, Jun 28 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..435
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Column k=5 of A289192.

Cf. A160609, A160610.

Programs

Magma

[(Factorial(n)*(&+[Binomial(n,k)*(5^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 09 2018

Maple

a:= n-> n! * add(binomial(n, i)*5^i/i!, i=0..n):
seq(a(n), n=0..20);

Mathematica

[Table[n!*LaguerreL[n,-5], {n,0,50}]] (* G. C. Greubel, May 09 2018 *)

PARI

for(n=0,30, print1(n!*sum(k=0,n, binomial(n,k)*(5^k/k!)), ", ")) \\ G. C. Greubel, May 09 2018

PARI

a(n) = n!*pollaguerre(n, 0, -5); \\ Michel Marcus, Feb 05 2021

Formula

E.g.f.: exp(5*x/(1-x))/(1-x).

a(n) = n! * Sum_{i=0..n} 5^i/i! * binomial(n,i).

a(n) = n! * A160609(n)/A160610(n).

a(n) ~ exp(-5/2 + 2*sqrt(5*n) - n) * n^(n + 1/4) / (sqrt(2)*5^(1/4)) * (1 + 223/(48*sqrt(5*n))). - Vaclav Kotesovec, Nov 13 2017

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 5^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020