%I #31 May 02 2022 08:14:34

%S 1,5,45,630,12600,340200,11907000,523908000,28291032000,1838917080000,

%T 141596615160000,12743695364400000,1325344317897600000,

%U 157715973829814400000,21291656467024944000000

%N a(n) = polygorial(n,3)/polygorial(3,n), n >= 3.

%H G. C. Greubel, <a href="/A085356/b085356.txt">Table of n, a(n) for n = 0..220</a>

%H Daniel Dockery, <a href="https://web.archive.org/web/20140617132401/http://danieldockery.com/res/math/polygorials.pdf">Polygorials, Special "Factorials" of Polygonal Numbers</a>, preprint, 2003.

%F a(n) = polygorial(n+3, 3)/polygorial(3, n+3) = (n+1)!^2*(n+2)*(n+3)*(n+4)/(2^n*24) = A067550(n+2)/2.

%F a(n) ~ (1/12)*Pi*n^(2*n+6)/(2^n*exp(2*n)). - _Ilya Gutkovskiy_, Dec 17 2016

%F D-finite with recurrence 2*a(n) = (n+4)*(n+1)*a(n-1). - _R. J. Mathar_, Mar 12 2019

%p a := n->(n+1)!^2*(n+2)*(n+3)*(n+4)/2^n/24; [seq(a(j),j=0..15)];

%p seq(mul(binomial(k,2)-binomial(k,1), k =5..n), n=4..18 ); # _Zerinvary Lajos_, Aug 07 2007

%t polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[ polygorial[3, #]/polygorial[#, 3] &, 17, 3] (* _Robert G. Wilson v_, Dec 13 2016 *)

%Y Cf. A067550, A084939, A084940, A084941, A084942, A084943, A084944.

%K easy,nonn

%O 0,2

%A Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003