A289213 a(n) = n! * Laguerre(n,-7).

1, 8, 79, 916, 12113, 179152, 2921911, 51988748, 1000578817, 20686611736, 456805020959, 10721879413252, 266382974861521, 6980304560060384, 192311632290456007, 5555079068684580988, 167822887344661475969, 5290815252203206305832, 173713426149927498289903

Offset: 0

Author

Alois P. Heinz, Jun 28 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..432
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Column k=7 of A289192.

Cf. A160607, A160608.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(7*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 13 2018

Maple

a:= n-> n! * add(binomial(n, i)*7^i/i!, i=0..n):
seq(a(n), n=0..20);

Mathematica

Table[n!*LaguerreL[n, -7], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)

PARI

x = 'x + O('x^30); Vec(serlaplace(exp(7*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017

Python

from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -7))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

Formula

E.g.f.: exp(7*x/(1-x))/(1-x).

a(n) = n! * Sum_{i=0..n} 7^i/i! * binomial(n,i).

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

a(n) ~ exp(-7/2 + 2*sqrt(7*n) - n) * n^(n + 1/4) / (sqrt(2)*7^(1/4)) * (1 + 367/(48*sqrt(7*n))). - Vaclav Kotesovec, Nov 13 2017

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 7^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020