A140702 Main diagonal of array A(k,n) = product of first n centered n-gonal numbers.
40, 1625, 151776, 27316471, 8429601664, 4108830350625, 2977546171600000, 3062351613203813051, 4308809606735976861696, 8050856986181775515023417, 19490752185922086291273856000, 59888297825402713913058605859375, 229474927848540723655596345639141376
Offset: 3
Examples
a(3) = 3rd centered polygorial number polygorial(3,3) = A140701(3) = product of the first 3 centered triangular numbers = 1 * 4 * 10 = 40. a(4) = 4th centered polygorial number centered polygorial(4,4) = product of the first 4 centered square numbers A001844 = 1 * 5 * 13 * 25 = 1625. 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. 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.
Links
- Nathaniel Johnston, Table of n, a(n) for n = 3..100
- Eric W. Weisstein, Centered Triangular Number.
Programs
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Maple
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
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Mathematica
Table[Product[n*k*(k-1)/2+1,{k,1,n}],{n,3,20}] (* Vaclav Kotesovec, Jul 11 2015 *)
Formula
a(n) ~ Pi * n^(3*n-1) / (exp(2*n) * 2^(n-2)). - Vaclav Kotesovec, Jul 11 2015
Extensions
a(9) corrected and more terms from Nathaniel Johnston, Oct 01 2011
Comments