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%I #15 Feb 11 2019 21:54:16
%S 40,1625,151776,27316471,8429601664,4108830350625,2977546171600000,
%T 3062351613203813051,4308809606735976861696,8050856986181775515023417,
%U 19490752185922086291273856000,59888297825402713913058605859375,229474927848540723655596345639141376
%N Main diagonal of array A(k,n) = product of first n centered n-gonal numbers.
%C For analog with regular (not centered) n-gonal numbers, see A133401.
%C Array A(k,n) = k-th polygorial(n,k) begins:
%C k | CenteredPolygorial(n,k)
%C ---+-------------------------
%C 3 | 1 4 40 760 23560 1083760 69360640 5895654400 A140701
%C 4 | 1 5 65 1625 66625 4064125 345450625 39035920625
%C 5 | 1 6 96 2976 151776 11534976 1222707456 172401751296
%C 6 | 1 7 133 4921 300181 27316471 3469191817 586293417073
%C 7 | 1 8 176 7568 537328 56956768 8429601664 1660631527808
%C 8 | 1 9 225 11025 893025 108056025 18261468225 4108830350625
%C 9 | 1 10 280 15400 1401400 190590400 36212176000 9161680528000
%H Nathaniel Johnston, <a href="/A140702/b140702.txt">Table of n, a(n) for n = 3..100</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/CenteredTriangularNumber.html">Centered Triangular Number</a>.
%F a(n) ~ Pi * n^(3*n-1) / (exp(2*n) * 2^(n-2)). - _Vaclav Kotesovec_, Jul 11 2015
%e a(3) = 3rd centered polygorial number polygorial(3,3) = A140701(3) = product of the first 3 centered triangular numbers = 1 * 4 * 10 = 40.
%e a(4) = 4th centered polygorial number centered polygorial(4,4) = product of the first 4 centered square numbers A001844 = 1 * 5 * 13 * 25 = 1625.
%e a(5) = 5th centered pentagorial number centered polygorial(5,5) = product of the first 5 centered pentagonal numbers A005891 = 1 * 5 * 12 * 22 * 35 = 151776.
%e a(6) = 6th centered hexagorial number centered polygorial(6,6) = product of the first 6 centered hexagonal numbers A003215 = 1 * 7 * 19 * 37 * 61 * 91 = 27316471.
%p A140702 := proc(n) mul(n*k*(k-1)/2+1,k=1..n): end: seq(A140702(n),n=3..15); # _Nathaniel Johnston_, Oct 01 2011
%t Table[Product[n*k*(k-1)/2+1,{k,1,n}],{n,3,20}] (* _Vaclav Kotesovec_, Jul 11 2015 *)
%Y Cf. A005448, A006003, A006472, A133401, A140701.
%K easy,nonn
%O 3,1
%A _Jonathan Vos Post_, May 24 2008
%E a(9) corrected and more terms from _Nathaniel Johnston_, Oct 01 2011