A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2.

18, 576, 46200, 7484400, 2137544640, 981562982400, 678245967907200, 670873729125600000, 913601739437346960000, 1660189302321994373529600, 3923769742187622047360640000, 11805614186177306251101945600000, 44403795869109177300313209696000000

Offset: 3

Author

Jonathan Vos Post, Nov 25 2007

Keywords

• easy
• nonn

Comments

Array T(n,k) = k-th polygorial(n,k) begins:

k | polygorial(n,k)

3 | 1 1 3 18 180 2700 56700 1587600 57153600

4 | 1 1 4 36 576 14400 518400 25401600 1625702400

5 | 1 1 5 60 1320 46200 2356200 164934000 15173928000

6 | 1 1 6 90 2520 113400 7484400 681080400 81729648000

7 | 1 1 7 126 4284 235620 19085220 2137544640 316356606720

8 | 1 1 8 168 6720 436800 41932800 5577062400 981562982400

9 | 1 1 9 216 9936 745200 82717200 12738448800 2598643555200

10 | 1 1 10 270 14040 1193400 150368400 26314470000 6104957040000

Examples

			a(3) = polygorial(3,3) = A006472(3) = product of the first 3 triangular numbers = 1*3*6 = 18.
a(4) = polygorial(4,4) = A001044(4) = product of the first 4 squares = 1*4*9*16 = 576.
a(5) = polygorial(5,5) = A084939(5) = product of the first 5 pentagonal numbers = 1*5*12*22*35 = 46200.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 3..100
• Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Crossrefs

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084943, A084944, A085356.

Programs

Maple

A133401 := proc(n) return mul((n/2-1)*m^2-(n/2-2)*m,m=1..n): end: seq(A133401(n),n=3..15); # Nathaniel Johnston, May 05 2011

Mathematica

Table[Product[m*(4 - n + m*(n-2))/2, {m, 1, n}],{n, 3, 20}] (* Vaclav Kotesovec, Feb 20 2015 *)
Table[FullSimplify[(n-2)^n * Gamma[n+1] * Gamma[n+2/(n-2)] / (2^n*Gamma[2/(n-2)])],{n,3,15}] (* Vaclav Kotesovec, Feb 20 2015 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k - 2), n]]; Array[ polygorial[#, #] &, 13, 3] (* Robert G. Wilson v, Dec 13 2016 *)

Formula

a(n) ~ Pi * n^(3*n-1) / (2^(n-2) * exp(2*n+2)). - Vaclav Kotesovec, Feb 20 2015

Extensions

Edited by Nathaniel Johnston, May 05 2011