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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 22 10:32 EST 2019. Contains 319363 sequences. (Running on oeis4.)