A289212 a(n) = n! * Laguerre(n,-6).

1, 7, 62, 654, 7944, 108696, 1649232, 27422352, 495057024, 9631281024, 200682406656, 4455296877312, 104921038236672, 2610989435003904, 68430995893131264, 1883330926998829056, 54286270223002140672, 1635031821385383247872, 51347572582353094508544

Offset: 0

Author

Alois P. Heinz, Jun 28 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..434
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Column k=6 of A289192.

Cf. A160607, A160608.

Programs

Maple

a:= n-> n! * add(binomial(n, i)*6^i/i!, i=0..n):
seq(a(n), n=0..20);

Mathematica

Table[n!*LaguerreL[n, -6], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)

PARI

my(x = 'x + O('x^30)); Vec(serlaplace(exp(6*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017

PARI

a(n) = n!*pollaguerre(n, 0, -6); \\ Michel Marcus, Feb 05 2021

Python

from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -6))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

Formula

E.g.f.: exp(6*x/(1-x))/(1-x).

a(n) = n! * Sum_{i=0..n} 6^i/i! * binomial(n,i).

a(n) = n! * A160607(n)/A160608(n).

a(n) ~ exp(-3 + 2*sqrt(6*n) - n) * n^(n + 1/4) / (2^(3/4)*3^(1/4)) * (1 + 97/(16*sqrt(6*n))). - Vaclav Kotesovec, Nov 13 2017

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 6^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020