A289215 a(n) = n! * Laguerre(n,-9).

1, 10, 119, 1626, 24945, 422994, 7836255, 157169826, 3388099329, 78031713690, 1910451937671, 49510386761130, 1353167691897969, 38878205830928226, 1170930069982659375, 36875214316479123954, 1211549306913066598785, 41445016025330141416746

Offset: 0

Author

Alois P. Heinz, Jun 28 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..430
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Column k=9 of A289192.

Cf. A160601, A160602.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(9*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 11 2018

Maple

a:= n-> n! * add(binomial(n, i)*9^i/i!, i=0..n):
seq(a(n), n=0..20);

Mathematica

Table[n!*LaguerreL[n, -9], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)

PARI

my(x = 'x + O('x^30)); Vec(serlaplace(exp(9*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017

PARI

a(n) = n!*pollaguerre(n, 0, -9); \\ Michel Marcus, Feb 05 2021

Python

from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -9))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

Formula

E.g.f.: exp(9*x/(1-x))/(1-x).

a(n) = n! * Sum_{i=0..n} 9^i/i! * binomial(n,i).

a(n) = n! * A160601(n)/A160602(n).

a(n) ~ exp(-9/2 + 6*sqrt(n) - n) * n^(n + 1/4) / sqrt(6) * (1 + 181/(48*sqrt(n))). - Vaclav Kotesovec, Nov 13 2017

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 9^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020