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A373208 Decimal expansion of Product_{k>=1} f(2*k)^2/(f(2*k-1) * f(2*k+1)), where f(k) = k^(1/k^2).

%I A373208 #6 May 28 2024 05:29:04
%S A373208 1,2,2,4,6,2,3,1,4,0,5,8,5,1,1,1,1,4,5,5,9,5,2,5,7,0,4,5,1,6,2,1,5,8,
%T A373208 9,4,7,2,0,1,0,1,8,4,4,8,3,2,0,3,2,1,5,1,9,8,3,1,0,8,8,2,7,8,9,9,0,7,
%U A373208 0,6,9,3,3,4,7,9,0,1,1,6,5,5,6,5,4,0,0,4,3,2,5,0,6,1,3,1,8,4,4,2,2,7,3,8,0
%N A373208 Decimal expansion of Product_{k>=1} f(2*k)^2/(f(2*k-1) * f(2*k+1)), where f(k) = k^(1/k^2).
%H A373208 Dirk Huylebrouck, <a href="https://doi.org/10.4169/amer.math.monthly.122.04.371">Generalizing Wallis' formula</a>, The American Mathematical Monthly, Vol. 122, No. 4 (2015), pp. 371-372; <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.122.04.371">alternative link</a>; <a href="https://arxiv.org/abs/1402.6577">arXiv preprint</a>, arXiv:1402.6577 [math.HO], 2014.
%H A373208 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet Eta Function</a>.
%H A373208 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>.
%F A373208 Equals exp(2*eta'(2)) = exp(2*A210593), where eta is the Dirichlet eta function.
%F A373208 Equals (4*Pi*exp(gamma)/A^12)^zeta(2), where gamma is Euler's constant (A001620) and A is the Glaisher-Kinkelin constant (A074962).
%e A373208 (2^(1/2^2)/1^1^2) * (2^(1/2^2)/3^(1/3^2)) * (4^(1/4^2)/3^(1/3^2)) * (4^(1/4^2)/5^(1/5^2)) * ...
%e A373208 1.22462314058511114559525704516215894720101844832032...
%t A373208 RealDigits[(4 * Pi * Exp[EulerGamma] / Glaisher^12)^Zeta[2], 10, 120][[1]]
%o A373208 (PARI) (4 * Pi * exp(Euler - 1 + 12*zeta'(-1)))^zeta(2)
%Y A373208 Cf. A001620, A013661, A073004, A074962, A210593, A373207.
%K A373208 nonn,cons
%O A373208 1,2
%A A373208 _Amiram Eldar_, May 28 2024