A334449 - OEIS

%I #17 Jun 27 2020 11:52:01

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

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

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

%N Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^5).

%C In general, for s>0, Product_{k>=1} (1 + 1/A002144(k)^(2*s+1))/(1 - 1/A002144(k)^(2*s+1)) = Pi^(2*s+1) * A000364(s) * zeta(2*s+1) / ((2^(2*s+2) + 2) * (2*s)! * zeta(4*s+2)). - _Dimitris Valianatos_, May 01 2020

%C In general, for s>1, Product_{k>=1} (1 + 1/A002144(k)^s)/(1 - 1/A002144(k)^s) = (zeta(s, 1/4) - zeta(s, 3/4)) * zeta(s) / (2^s * (2^s + 1) * zeta(2*s)).

%D B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

%H Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta function expansions of some classical constants</a>, Feb 18 1996, p. 7-8.

%F A334449 / A334450 = 4725*zeta(5)/(16*Pi^5).

%F A334449 * A334451 = 90720*zeta(5)/Pi^10.

%e 1.0003234751480716386036864273399423692652465522027379804075071648599638113746...

%Y Cf. A002144, A243380, A334424, A334445.

%K nonn,cons

%O 1,5

%A _Vaclav Kotesovec_, Apr 30 2020

%E More digits from _Vaclav Kotesovec_, Jun 27 2020