A062141 Third column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

1, 18, 252, 3360, 45360, 635040, 9313920, 143700480, 2335132800, 39956716800, 719220902400, 13599813427200, 269729632972800, 5602076992512000, 121645100408832000, 2757288942600192000, 65140951268929536000, 1601701037083090944000, 40932359836567879680000

Offset

0,2

Links

Indranil Ghosh, Table of n, a(n) for n = 0..400

Index entries for sequences related to Laguerre polynomials

Formula

a(n) = (n+2)!*binomial(n+5, 5)/2!.

E.g.f.: (1 + 10*x + 10*x^2)/(1-x)^8.

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-5) = (-1)^(n-1)*f(n,5,-3), (n >= 5). - Milan Janjic, Mar 01 2009

Example

a(2) = (2+2)! * binomial(2+5,5) / 2! = (24*21)/2 = 252. - Indranil Ghosh, Feb 24 2017

Maple

a:= n->(n+2)!*binomial(n+5, 5)/2!: seq(a(n), n=0..20); # Zerinvary Lajos, Apr 29 2007

Mathematica

Table[(n+2)!*Binomial[n+5, 5]/2!, {n, 0, 15}] (* Indranil Ghosh, Feb 24 2017 *)

Prog

(Sage) [binomial(n, 5)*factorial (n-3)/2 for n in range(5, 21)] # Zerinvary Lajos, Jul 07 2009
(PARI) a(n)=(n+2)!*binomial(n+5, 5)/2! \\ Indranil Ghosh, Feb 24 2017
(Python)
import math
f=math.factorial
def C(n, r):return f(n)/f(r)/f(n-r)
def A062141(n): return f(n+2)*C(n+5, 5)/f(2) # Indranil Ghosh, Feb 24 2017
(PARI) x='x+O('x^30); Vec(serlaplace((1+10*x+10*x^2)/(1-x)^8)) \\ G. C. Greubel, May 11 2018
(Magma) [Factorial(n+2)*Binomial(n+5, 5)/2: n in [0..20]]; // G. C. Greubel, May 11 2018

Crossrefs

Cf. A061206, A062137.

Sequence in context: A125475 A255371 A016175 * A157708 A159537 A136660

Adjacent sequences: A062138 A062139 A062140 * A062142 A062143 A062144

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 19 2001

Status

approved