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%I A133401 #27 Sep 04 2019 10:52:34
%S A133401 18,576,46200,7484400,2137544640,981562982400,678245967907200,
%T A133401 670873729125600000,913601739437346960000,1660189302321994373529600,
%U A133401 3923769742187622047360640000,11805614186177306251101945600000,44403795869109177300313209696000000
%N A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2.
%C A133401 Array T(n,k) = k-th polygorial(n,k) begins:
%C A133401 k  |  polygorial(n,k)
%C A133401 3  | 1 1  3  18   180    2700     56700     1587600      57153600
%C A133401 4  | 1 1  4  36   576   14400    518400    25401600    1625702400
%C A133401 5  | 1 1  5  60  1320   46200   2356200   164934000   15173928000
%C A133401 6  | 1 1  6  90  2520  113400   7484400   681080400   81729648000
%C A133401 7  | 1 1  7 126  4284  235620  19085220  2137544640  316356606720
%C A133401 8  | 1 1  8 168  6720  436800  41932800  5577062400  981562982400
%C A133401 9  | 1 1  9 216  9936  745200  82717200 12738448800 2598643555200
%C A133401 10 | 1 1 10 270 14040 1193400 150368400 26314470000 6104957040000
%H A133401 Nathaniel Johnston, <a href="/A133401/b133401.txt">Table of n, a(n) for n = 3..100</a>
%H A133401 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 A133401 a(n) ~ Pi * n^(3*n-1) / (2^(n-2) * exp(2*n+2)). - _Vaclav Kotesovec_, Feb 20 2015
%e A133401 a(3) = polygorial(3,3) = A006472(3) = product of the first 3 triangular numbers = 1*3*6 = 18.
%e A133401 a(4) = polygorial(4,4) = A001044(4) = product of the first 4 squares = 1*4*9*16 = 576.
%e A133401 a(5) = polygorial(5,5) = A084939(5) = product of the first 5 pentagonal numbers = 1*5*12*22*35 = 46200.
%p A133401 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
%t A133401 Table[Product[m*(4 - n + m*(n-2))/2, {m, 1, n}],{n, 3, 20}] (* _Vaclav Kotesovec_, Feb 20 2015 *)
%t A133401 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 *)
%t A133401 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 *)
%Y A133401 Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084943, A084944, A085356.
%K A133401 easy,nonn
%O A133401 3,1
%A A133401 _Jonathan Vos Post_, Nov 25 2007
%E A133401 Edited by _Nathaniel Johnston_, May 05 2011

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