A062143 Fifth column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

1, 40, 1080, 25200, 554400, 11975040, 259459200, 5708102400, 128432304000, 2968213248000, 70643475302400, 1733976211968000, 43927397369856000, 1148870392750080000, 31019500604252160000, 864410083505160192000

Offset: 0

Author

Wolfdieter Lang, Jun 19 2001

Keywords

• nonn
• easy

Comments

The coefficients of the numerator polynomials N(m,x) of the e.g.f. for column m (here m=4) give triangle A062145.

Examples

			a(3) = (3+4)! * binomial(3+7,7) / 4! = (5040 * 120) / 24 = 25200. - _Indranil Ghosh_, Feb 23 2017
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..400
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A062137, A062142, A062145.

Programs

Magma

[Factorial(n+4)*Binomial(n+7,7)/Factorial(4): n in [0..20]]; // G. C. Greubel, May 12 2018

Mathematica

Table[(n+4)!*Binomial[n+7,7]/4!,{n,0,15}] (* Indranil Ghosh, Feb 23 2017 *)

PARI

a(n) = (n+4)!*binomial(n+7,7)/4! \\ Indranil Ghosh, Feb 23 2017

Python

import math
f=math.factorial
def C(n,r):return f(n)/f(r)/f(n-r)
def A062143(n):return f(n+4)*C(n+7,7)/f(4) # Indranil Ghosh, Feb 23 2017

Formula

a(n) = (n+4)!*binomial(n+7, 7)/4!;

E.g.f.: (1 + 28*x + 126*x^2 + 140*x^3 + 35*x^4)/(1-x)^12.

If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-4) = (-1)^n*f(n,4,-8), (n>=4). - Milan Janjic, Mar 01 2009