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A140702 Main diagonal of array A(k,n) = product of first n centered n-gonal numbers. 2
40, 1625, 151776, 27316471, 8429601664, 4108830350625, 2977546171600000, 3062351613203813051, 4308809606735976861696, 8050856986181775515023417, 19490752185922086291273856000, 59888297825402713913058605859375, 229474927848540723655596345639141376 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 19 12:48 EST 2020. Contains 331049 sequences. (Running on oeis4.)