A156648 Decimal expansion of Product_{k>=1} (1 + 1/k^2).

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

Offset: 1

Author

R. J. Mathar, Feb 12 2009

Keywords

• cons
• easy
• nonn

Comments

Consider the value at s = 2 of the partition zeta functions zeta_{type}(s), where the defining sum runs over partitions into 'type' parts, where 'type' is 'even', 'prime' or 'distinct'. (For the precise definitions see R. Schneider's dissertation.) Then

zeta_{even}(2) = Pi/2 = A019669;

zeta_{prime}(2) = Pi^2/6 = A013661;

zeta_{distinct}(2) = sinh(Pi)/Pi, this constant. - Peter Luschny, Aug 11 2021

For m>0, Product_{k>=1} (1 + m/k^2) = sinh(Pi*sqrt(m)) / (Pi*sqrt(m)). - Vaclav Kotesovec, Aug 30 2024

Examples

			3.676077910374977720695697492028260666507156346827630277478003593557447324111... = (1+1)*(1+1/4)*(1+1/9)*(1+1/16)*(1+1/25)*...
		

References

• Reinhold Remmert, Classical topics in complex function theory, Vol. 172 of Graduate Texts in Mathematics, p. 12, Springer, 1997.

Links

• Robert Schneider, Eulerian series, zeta functions and the arithmetic of partitions, arXiv:2008.04243 [math.NT], 2020.

Crossrefs

Square root of A084243.

Cf. A084243, A073017, A258870, A258871, A334411, A019669, A013661.

Programs

Maple

evalf(sinh(Pi)/Pi) ;

Mathematica

RealDigits[Sinh[Pi]/Pi, 10, 111][[1]] (* or *)
RealDigits[Re[1/(I!*(-I)!)], 10, 111][[1]] (* Robert G. Wilson v, Feb 23 2015 *)

PARI

sinh(Pi)/Pi \\ Charles R Greathouse IV, Dec 16 2013

PARI

prodnumrat(1 + 1/x^2, 1) \\ Charles R Greathouse IV, Feb 03 2025

Formula

Equals sinh(Pi)/Pi.

Equals 1/A090986. - R. J. Mathar, Mar 05 2009

Binomial(2, 1+i) = 1/(i!*(-i)!) (where x! means Gamma(x+1)). - Robert G. Wilson v, Feb 23 2015

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(2*j)/j)). - Vaclav Kotesovec, Mar 28 2019

Equals Product_{k>=1} (1+2/(k*(k+2))). - Amiram Eldar, Aug 16 2020