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Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^4).
6

%I #13 Jun 27 2020 11:52:37

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

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

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

%N Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^4).

%C In general, for s>1, Product_{k>=1} (1 + 1/A002145(k)^s)/(1 - 1/A002145(k)^s) = 2^s * (2^s - 1) * zeta(s) / (zeta(s, 1/4) - zeta(s, 3/4)) = 1 / (2 * (-1)^s * PolyGamma(s-1, 1/4) / (2^s * (2^s - 1) * Gamma(s) * zeta(s)) - 1).

%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 A334447 / A334448 = 1/(PolyGamma(3, 1/4)/(8*Pi^4) - 1).

%F A334445 * A334447 = 1680 / (17*Pi^4).

%e 1.01284973750365824105373880963011203968450421655386945092221...

%Y Cf. A002145, A243381, A334426, A334451.

%K nonn,cons

%O 1,4

%A _Vaclav Kotesovec_, Apr 30 2020

%E More digits from _Vaclav Kotesovec_, Jun 27 2020