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).

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, 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, 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

Offset

1,2

Links

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

Dirk Huylebrouck, Generalizing Wallis' formula, The American Mathematical Monthly, Vol. 122, No. 4 (2015), pp. 371-372; alternative link; arXiv preprint, arXiv:1402.6577 [math.HO], 2014.

Eric Weisstein's World of Mathematics, Dirichlet Eta Function.

Wikipedia, Dirichlet eta function.

Formula

Equals exp(2*eta'(2)) = exp(2*A210593), where eta is the Dirichlet eta function.

Equals (4*Pi*exp(gamma)/A^12)^zeta(2), where gamma is Euler's constant (A001620) and A is the Glaisher-Kinkelin constant (A074962).

Example

(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)) * ...
1.22462314058511114559525704516215894720101844832032...

Mathematica

RealDigits[(4 * Pi * Exp[EulerGamma] / Glaisher^12)^Zeta[2], 10, 120][[1]]

Prog

(PARI) (4 * Pi * exp(Euler - 1 + 12*zeta'(-1)))^zeta(2)

Crossrefs

Cf. A001620, A013661, A073004, A074962, A210593, A373207.

Sequence in context: A019964 A087377 A010768 * A143267 A113463 A278217

Adjacent sequences: A373205 A373206 A373207 * A373209 A373210 A373211

Keyword

nonn,cons

Author

Amiram Eldar, May 28 2024

Status

approved