A085356 a(n) = polygorial(n,3)/polygorial(3,n), n >= 3.

1, 5, 45, 630, 12600, 340200, 11907000, 523908000, 28291032000, 1838917080000, 141596615160000, 12743695364400000, 1325344317897600000, 157715973829814400000, 21291656467024944000000

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..220
• Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Crossrefs

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

Programs

Maple

a := n->(n+1)!^2*(n+2)*(n+3)*(n+4)/2^n/24; [seq(a(j),j=0..15)];
seq(mul(binomial(k,2)-binomial(k,1), k =5..n), n=4..18 ); # Zerinvary Lajos, Aug 07 2007

Mathematica

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 *)

Formula

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.

a(n) ~ (1/12)*Pi*n^(2*n+6)/(2^n*exp(2*n)). - Ilya Gutkovskiy, Dec 17 2016

D-finite with recurrence 2*a(n) = (n+4)*(n+1)*a(n-1). - R. J. Mathar, Mar 12 2019