A109219 Decimal expansion of Product_{n >= 2} 1-n^(-3).

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

Offset

0,1

Comments

The physical applications of this kind of product (with s<0) can be found in the Klauder et al. reference. - Karol A. Penson, Feb 24 2006

Links

Table of n, a(n) for n=0..104.

J. R. Klauder, K. A. Penson and J.-M. Sixdeniers, Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems, Physical Review A, Vol. 64, p. 013817 (2001).

Formula

Equals cosh((sqrt(3)*Pi)/2)/(3*Pi).

Product_{n >= 2} (1 - 1/n^p) simplifies, if p is odd, to 1/(p * Product(Gamma(- (-1)^(j*(1 + 1/p))), (j, 1, p - 1})) and, if p is even, to the elementary Product(Sin(Pi*(-1)^((2*j)/p))/(Pi*I), (j, 1, p/2 - 1}) / p. - David W. Cantrell, Feb 24 2006

Equals exp(Sum_{j>=1} (1 - zeta(3*j))/j). - Vaclav Kotesovec, Apr 27 2020

Equals 1/(Gamma((5-i*sqrt(3))/2)*Gamma((5+i*sqrt(3))/2)). - Amiram Eldar, Sep 01 2020

Example

0.809396597366290109578680478726382119372787648261130...

Mathematica

RealDigits[N[Cosh[(Sqrt[3]*Pi)/2]/(3*Pi), 150]] (* T. D. Noe, Apr 24 2006 *)

Prog

(PARI) exp(suminf(j=1, (1 - zeta(3*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020

Crossrefs

Cf. A073017, A175615, A175617, A175619, A258870.

Sequence in context: A198820 A138285 A085291 * A199506 A154803 A254178

Adjacent sequences: A109216 A109217 A109218 * A109220 A109221 A109222

Keyword

nonn,cons

Author

Zak Seidov, Apr 17 2006

Extensions

Corrected and extended by T. D. Noe, Apr 24 2006

Status

approved