A230821 - OEIS

%I #20 Jul 30 2023 02:25:17

%S 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,

%T 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,

%U 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

%N Decimal expansion of Product_{n>=3} cosh(Pi/n).

%F From _Amiram Eldar_, Jul 30 2023: (Start)

%F 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.

%F 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)

%e 6.15018022079681769526817395848212318982616780620281236328208157244...

%p evalf(product(1/sech(Pi/k), k=3..infinity), 120) # _Vaclav Kotesovec_, Sep 20 2014

%t (* 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]

%t 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 *)

%o (PARI) default(realprecision,150); exp(sumpos(n=3, log(cosh(Pi/n)))) \\ _Vaclav Kotesovec_, Sep 20 2014

%Y Cf. A051762.

%Y Cf. A027641, A027642.

%K nonn,easy,cons

%O 1,1

%A _Robert G. Wilson v_, Oct 30 2013

%E Corrected by _Vaclav Kotesovec_, Sep 20 2014