A373207 Decimal expansion of Product_{k>=1} f(2*k)^2/(f(2*k-1) * f(2*k+1)), where f(k) = k^(1/k).

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

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'(1)) = exp(2*A091812), where eta is the Dirichlet eta function.

Equals 2^(2*gamma - log(2)), where gamma is Euler's constant (A001620).

Example

(2^(1/2)/1^1) * (2^(1/2)/3^(1/3)) * (4^(1/4)/3^(1/3)) * (4^(1/4)/5^(1/5)) * ...
1.37676673907488822612716594825041629908712439037992...

Mathematica

RealDigits[2^(2*EulerGamma - Log[2]), 10, 120][[1]]

Prog

(PARI) 2^(2*Euler - log(2))

Crossrefs

Cf. A001620, A002162, A091812, A373208.

Sequence in context: A075785 A105735 A213610 * A296343 A347155 A371240

Adjacent sequences: A373204 A373205 A373206 * A373208 A373209 A373210

Keyword

nonn,cons

Author

Amiram Eldar, May 28 2024

Status

approved