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%I #35 Apr 22 2024 07:32:22
%S 1,6,144,8640,1036800,217728000,73156608000,36870930432000,
%T 26547069911040000,26281599211929600000,34691710959747072000000,
%U 59530976006925975552000000,130015651599126330605568000000,354942728865614882553200640000000,1192607568988466005378754150400000000
%N a(n) = (n+2)! * (n+1)! * n! / 2.
%H Harvey P. Dale, <a href="/A090443/b090443.txt">Table of n, a(n) for n = 0..180</a>
%H J. Agapito, <a href="https://dx.doi.org/10.1016/j.laa.2014.03.018">On symmetric polynomials with only real zeros and nonnegative gamma-vectors</a>, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
%H César Aguilera, <a href="https://hal.archives-ouvertes.fr/hal-02297262">On Zeta Functions and Tetrahedral Numbers</a>, hal-02297262 (2019) [math.NT].
%F Fourth column (m=3) of triangle A090441.
%F From _Karol A. Penson_ Jul 25 2013: (Start)
%F G.f. of hypergeometric type:
%F Sum_{n>=0} a(n)*z^n/(n!)^3 = (1+2*z)/(1-z)^4;
%F 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:
%F 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.
%F This solution of the Stieltjes moment problem is not unique.
%F 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)
%p a:=n->mul(j^3-j, j=2..n): seq(a(n), n=1..13); # _Zerinvary Lajos_, May 08 2008
%t (Times@@#)/2&/@Partition[Range[0,20]!,3,1] (* _Harvey P. Dale_, Dec 03 2017 *)
%o (Python)
%o from math import factorial
%o def A090443(n): return factorial(n)**3*(n+2)*(n+1)**2>>1 # _Chai Wah Wu_, Apr 22 2024
%Y Cf. A010790, A090444, A224900, A172492.
%K nonn,easy,changed
%O 0,2
%A _Wolfdieter Lang_, Dec 23 2003
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