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

1,1

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

References

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

Links

Table of n, a(n) for n=1..105.

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

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

Example

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

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 *)

Prog

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

Crossrefs

Square root of A084243.

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

Sequence in context: A245220 A165952 A306658 * A278688 A016616 A256936

Adjacent sequences: A156645 A156646 A156647 * A156649 A156650 A156651

Keyword

cons,easy,nonn

Author

R. J. Mathar, Feb 12 2009

Status

approved