A156648 - OEIS

%I #40 Feb 03 2025 09:55:20

%S 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,

%T 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,

%U 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

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

%C 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

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

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

%C   zeta_{distinct}(2) = sinh(Pi)/Pi, this constant. - _Peter Luschny_, Aug 11 2021

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

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

%H Robert Schneider, <a href="https://arxiv.org/abs/2008.04243">Eulerian series, zeta functions and the arithmetic of partitions</a>, arXiv:2008.04243 [math.NT], 2020.

%F Equals sinh(Pi)/Pi.

%F Equals 1/A090986. - _R. J. Mathar_, Mar 05 2009

%F Binomial(2, 1+i) = 1/(i!*(-i)!) (where x! means Gamma(x+1)). - _Robert G. Wilson v_, Feb 23 2015

%F Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(2*j)/j)). - _Vaclav Kotesovec_, Mar 28 2019

%F Equals Product_{k>=1} (1+2/(k*(k+2))). - _Amiram Eldar_, Aug 16 2020

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

%p evalf(sinh(Pi)/Pi) ;

%t RealDigits[Sinh[Pi]/Pi, 10, 111][[1]] (* or *)

%t RealDigits[Re[1/(I!*(-I)!)], 10, 111][[1]] (* _Robert G. Wilson v_, Feb 23 2015 *)

%o (PARI) sinh(Pi)/Pi \\ _Charles R Greathouse IV_, Dec 16 2013

%o (PARI) prodnumrat(1 + 1/x^2, 1) \\ _Charles R Greathouse IV_, Feb 03 2025

%Y Square root of A084243.

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

%K cons,easy,nonn,changed

%O 1,1

%A _R. J. Mathar_, Feb 12 2009