A146481 Decimal expansion of Product_{n>=2} (1 - 1/(n*(n-1))).

2, 9, 6, 6, 7, 5, 1, 3, 4, 7, 4, 3, 5, 9, 1, 0, 3, 4, 5, 7, 0, 1, 5, 5, 0, 2, 0, 2, 1, 9, 1, 4, 2, 8, 6, 4, 8, 6, 4, 8, 3, 1, 5, 1, 9, 1, 7, 8, 9, 4, 7, 8, 9, 0, 8, 1, 6, 7, 3, 5, 7, 3, 3, 1, 6, 5, 9, 0, 6, 1, 6, 2, 9, 1, 5, 1, 9, 6, 0, 8, 8, 8, 3, 6, 6, 7, 4, 8, 1, 6, 4, 0, 2, 1, 2, 6, 2, 2, 1, 4, 5, 4, 1, 7, 7

Offset: 0

Author

R. J. Mathar, Feb 13 2009

Keywords

• nonn
• cons

Comments

Product of Artin's constant A005596 and the equivalent almost-prime products.

Examples

			0.2966751347435910345... = (1 - 1/2)*(1 - 1/6)*(1 - 1/12)*(1 - 1/20)*...
		

Links

• M. Chamberland, A. Straub, On gamma constants and infinite products, arXiv:1309.3455
• R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], first line Table 3.

Crossrefs

Cf. A005596.

Programs

Maple

phi := (1+sqrt(5))/2; evalf(-sin(Pi*phi)/Pi) ; # R. J. Mathar, Feb 20 2009

Mathematica

RealDigits[-Cos[Pi*Sqrt[5]/2]/Pi, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

Formula

The logarithm is -Sum_{s>=2} Sum_{j=1..floor(s/(1+r))} binomial(s-r*j-1, j-1)*(1-Zeta(s))/j at r=1.

s*Sum_{j=1..floor(s/2)} binomial(s-j-1, j-1)/j = A001610(s-1).

Equals 1/Product_{k=1..2} Gamma(1-x_k) = -sin(A094886)/A000796, where x_k are the 2 roots of the polynomial x*(x+1)-1. [R. J. Mathar, Feb 20 2009]

Extensions

More terms from Jean-François Alcover, Feb 11 2013