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A140702 Main diagonal of array A(k,n) = product of first n centered n-gonal numbers.

%I A140702 #16 Feb 16 2025 08:33:08
%S A140702 40,1625,151776,27316471,8429601664,4108830350625,2977546171600000,
%T A140702 3062351613203813051,4308809606735976861696,8050856986181775515023417,
%U A140702 19490752185922086291273856000,59888297825402713913058605859375,229474927848540723655596345639141376
%N A140702 Main diagonal of array A(k,n) = product of first n centered n-gonal numbers.
%C A140702 For analog with regular (not centered) n-gonal numbers, see A133401.
%C A140702 Array A(k,n) = k-th polygorial(n,k) begins:
%C A140702 k  |  CenteredPolygorial(n,k)
%C A140702 ---+-------------------------
%C A140702 3  | 1 4   40   760   23560    1083760      69360640      5895654400   A140701
%C A140702 4  | 1 5   65  1625   66625    4064125     345450625     39035920625
%C A140702 5  | 1 6   96  2976   151776   11534976   1222707456    172401751296
%C A140702 6  | 1 7  133  4921   300181   27316471   3469191817    586293417073
%C A140702 7  | 1 8  176  7568   537328   56956768   8429601664   1660631527808
%C A140702 8  | 1 9  225 11025   893025  108056025  18261468225   4108830350625
%C A140702 9  | 1 10 280 15400  1401400  190590400  36212176000   9161680528000
%H A140702 Nathaniel Johnston, <a href="/A140702/b140702.txt">Table of n, a(n) for n = 3..100</a>
%H A140702 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/CenteredTriangularNumber.html">Centered Triangular Number</a>.
%F A140702 a(n) ~ Pi * n^(3*n-1) / (exp(2*n) * 2^(n-2)). - _Vaclav Kotesovec_, Jul 11 2015
%e A140702 a(3) = 3rd centered polygorial number polygorial(3,3) = A140701(3) = product of the first 3 centered triangular numbers = 1 * 4 * 10 = 40.
%e A140702 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 A140702 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 A140702 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 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 A140702 Table[Product[n*k*(k-1)/2+1,{k,1,n}],{n,3,20}] (* _Vaclav Kotesovec_, Jul 11 2015 *)
%Y A140702 Cf. A005448, A006003, A006472, A133401, A140701.
%K A140702 easy,nonn
%O A140702 3,1
%A A140702 _Jonathan Vos Post_, May 24 2008
%E A140702 a(9) corrected and more terms from _Nathaniel Johnston_, Oct 01 2011