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

3,1

Comments

For analog with regular (not centered) n-gonal numbers, see A133401.

Array A(k,n) = k-th polygorial(n,k) begins:

k | CenteredPolygorial(n,k)

---+-------------------------

3 | 1 4 40 760 23560 1083760 69360640 5895654400 A140701

4 | 1 5 65 1625 66625 4064125 345450625 39035920625

5 | 1 6 96 2976 151776 11534976 1222707456 172401751296

6 | 1 7 133 4921 300181 27316471 3469191817 586293417073

7 | 1 8 176 7568 537328 56956768 8429601664 1660631527808

8 | 1 9 225 11025 893025 108056025 18261468225 4108830350625

9 | 1 10 280 15400 1401400 190590400 36212176000 9161680528000

Links

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

Eric W. Weisstein, Centered Triangular Number.

Formula

a(n) ~ Pi * n^(3*n-1) / (exp(2*n) * 2^(n-2)). - Vaclav Kotesovec, Jul 11 2015

Example

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.

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

Mathematica

Table[Product[n*k*(k-1)/2+1, {k, 1, n}], {n, 3, 20}] (* Vaclav Kotesovec, Jul 11 2015 *)

Crossrefs

Cf. A005448, A006003, A006472, A133401, A140701.

Sequence in context: A229635 A278431 A229584 * A223609 A145294 A147520

Adjacent sequences: A140699 A140700 A140701 * A140703 A140704 A140705

Keyword

easy,nonn

Author

Jonathan Vos Post, May 24 2008

Extensions

a(9) corrected and more terms from Nathaniel Johnston, Oct 01 2011

Status

approved