A230821 Decimal expansion of Product_{n>=3} cosh(Pi/n).
6, 1, 5, 0, 1, 8, 0, 2, 2, 0, 7, 9, 6, 8, 1, 7, 6, 9, 5, 2, 6, 8, 1, 7, 3, 9, 5, 8, 4, 8, 2, 1, 2, 3, 1, 8, 9, 8, 2, 6, 1, 6, 7, 8, 0, 6, 2, 0, 2, 8, 1, 2, 3, 6, 3, 2, 8, 2, 0, 8, 1, 5, 7, 2, 4, 4, 0, 2, 5, 7, 8, 8, 2, 8, 2, 2, 8, 7, 8, 3, 8, 9, 6, 3, 2, 4, 8, 9, 7, 6, 9, 7, 8, 4, 0, 5, 3, 2, 5, 9, 5, 7, 1, 4, 2
Offset: 1
Examples
6.15018022079681769526817395848212318982616780620281236328208157244...
Programs
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Maple
evalf(product(1/sech(Pi/k), k=3..infinity), 120) # Vaclav Kotesovec, Sep 20 2014
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Mathematica
(* RealDigits[ N[ Product[ Cosh[ Pi/n], {n, 3, Infinity}], 111]][[1]] *) [This approach turns out to give incorrect numerical results. - Vaclav Kotesovec, Sep 20 2014] Block[{$MaxExtraPrecision = 1000}, Do[Print[N[1/Exp[Sum[(-1)^(n+1) * (2^(2*n)-1)/n * Zeta[2*n]*(Zeta[2*n] - 1 - 1/2^(2*n)), {n, 1, m}]], 110]], {m, 250, 300}]] (* Vaclav Kotesovec, Sep 20 2014 *) -
PARI
default(realprecision,150); exp(sumpos(n=3, log(cosh(Pi/n)))) \\ Vaclav Kotesovec, Sep 20 2014
Formula
From Amiram Eldar, Jul 30 2023: (Start)
Equals exp(Sum_{k>=1} (2^(2*k)-1)*B(2*k)*(2*Pi)^(2*k)*(zeta(2*k)-1-1/2^(2*k))/(2*k*(2*k)!)), where B(k) is the k-th Bernoulli number.
Equals exp(Sum_{k>=1} (-1)^(k+1)*(2^(2*k)-1)*zeta(2*k)*(zeta(2*k)-1-1/2^(2*k))/k). (End)
Extensions
Corrected by Vaclav Kotesovec, Sep 20 2014