A090443 a(n) = (n+2)! * (n+1)! * n! / 2.

1, 6, 144, 8640, 1036800, 217728000, 73156608000, 36870930432000, 26547069911040000, 26281599211929600000, 34691710959747072000000, 59530976006925975552000000, 130015651599126330605568000000, 354942728865614882553200640000000, 1192607568988466005378754150400000000

Offset

0,2

Links

Harvey P. Dale, Table of n, a(n) for n = 0..180

J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.

César Aguilera, On Zeta Functions and Tetrahedral Numbers, hal-02297262 (2019) [math.NT].

Formula

Fourth column (m=3) of triangle A090441.

From Karol A. Penson Jul 25 2013: (Start)

G.f. of hypergeometric type:

Sum_{n>=0} a(n)*z^n/(n!)^3 = (1+2*z)/(1-z)^4;

integral representation as n-th moment of a positive function w(x) on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation:

a(n) = int(x^n*w(x),x=0..infinity), n>=0 where w(x)=MeijerG([[],[]],[[2,1,0]],[]],x)/2, w(0)=1/2, limit(w(x),x=infinity)=0. w(x) is monotonically decreasing over (0,infinity). The Meijer G function above cannot be represented by any other known special function.

This solution of the Stieltjes moment problem is not unique.

Asymptotics: a(n)->(1/32)*Pi^(3/2)*sqrt(2)*(32*n^2+136*n+193)*exp(-3*n)*(n)^(5/2+3*n), for n->infinity. (End)

Maple

a:=n->mul(j^3-j, j=2..n): seq(a(n), n=1..13); # Zerinvary Lajos, May 08 2008

Mathematica

(Times@@#)/2&/@Partition[Range[0, 20]!, 3, 1] (* Harvey P. Dale, Dec 03 2017 *)

Prog

(Python)
from math import factorial
def A090443(n): return factorial(n)**3*(n+2)*(n+1)**2>>1 # Chai Wah Wu, Apr 22 2024

Crossrefs

Cf. A010790, A090444, A224900, A172492.

Sequence in context: A270504 A085905 A203978 * A307416 A133460 A166953

Adjacent sequences: A090440 A090441 A090442 * A090444 A090445 A090446

Keyword

nonn,easy,changed

Author

Wolfdieter Lang, Dec 23 2003

Status

approved