A140701 Partial products of A005448.

1, 1, 4, 40, 760, 23560, 1083760, 69360640, 5895654400, 642626329600, 87397180825600, 14507932017049600, 2887078471392870400, 678463440777324544000, 185898982772986925056000, 58744078556263868317696000, 21206612358811256462688256000

Offset: 0

Author

Jonathan Vos Post, May 24 2008

Keywords

• nonn
• easy

Examples

			a(10) = 87397180825600 = 1 * 4 * 10 * 19 * 31 * 46 * 64 * 85 * 109 * 136.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..244 [a(0) added by Robert C. Lyons].
• Eric Weisstein's World of Mathematics, Centered Triangular Number

Crossrefs

Cf. A005448.

For analog with centered n-gonal numbers see A140702.

For analog with regular triangular numbers see A006472.

For the analog with a partial sum instead of a partial product see A006003.

Programs

Mathematica

Table[Product[(3*k^2-3*k+2)/2,{k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Jul 11 2015 *)
FoldList[Times,3*Accumulate[Range[0,20]]+1] (* Harvey P. Dale, Aug 05 2018 *)

PARI

a(n) = prod(k=1, n, 3*k*(k-1)/2 + 1); \\ Michel Marcus, Mar 02 2023

Formula

a(n) = Product_{k=1..n} A005448(k).

a(n) ~ cosh(Pi*sqrt(5/3)/2) * 3^n * n^(2*n) / (exp(2*n) * 2^(n-1)). - Vaclav Kotesovec, Jul 11 2015

a(n) = (2/3)^(1 - n) * Pochhammer(1 + (3 - i*sqrt(15))/6, n - 1) * Pochhammer(1 + (3 + i*sqrt(15))/6, n - 1), for n>=1. - Antonio Graciá Llorente, Sep 10 2023