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A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2. 2
18, 576, 46200, 7484400, 2137544640, 981562982400, 678245967907200, 670873729125600000, 913601739437346960000, 1660189302321994373529600, 3923769742187622047360640000, 11805614186177306251101945600000, 44403795869109177300313209696000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

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

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 3..100

Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

FORMULA

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

EXAMPLE

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.

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 *)

CROSSREFS

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

Sequence in context: A183498 A254381 A177098 * A211708 A253826 A061079

Adjacent sequences:  A133398 A133399 A133400 * A133402 A133403 A133404

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Nov 25 2007

EXTENSIONS

Edited by Nathaniel Johnston, May 05 2011

STATUS

approved

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Last modified May 31 22:21 EDT 2020. Contains 334756 sequences. (Running on oeis4.)