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
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].
Programs
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Maple
a:=n->mul(j^3-j, j=2..n): seq(a(n), n=1..13); # Zerinvary Lajos, May 08 2008
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Mathematica
(Times@@#)/2&/@Partition[Range[0,20]!,3,1] (* Harvey P. Dale, Dec 03 2017 *)
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Python
from math import factorial def A090443(n): return factorial(n)**3*(n+2)*(n+1)**2>>1 # Chai Wah Wu, Apr 22 2024
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)
Comments