OFFSET
0,1
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
FORMULA
Equals exp(-Sum_{n>=1} (2^(2*n)-1) * Zeta(2*n)^2 / (n*Pi^(2*n)) ). - Vaclav Kotesovec, Sep 20 2014
Equals exp(Sum_{k>=1} (-1)^k*2^(2*k-1)*(2^(2*k)-1)*B(2*k)*zeta(2*k)/(k*(2*k)!)), where B(k) is the k-th Bernoulli number. - Amiram Eldar, Jul 30 2023
EXAMPLE
0.38853615333517585918432957568703590501390...
MAPLE
nn:= 120:
p:= product(cos(1/n), n=1..infinity):
f:= evalf(p, nn+10):
s:= convert(f, string):
seq(parse(s[n+1]), n=1..nn); # Alois P. Heinz, Nov 04 2013
MATHEMATICA
S = Series[Log[Cos[x]], {x, 0, 400}]; N[Exp[N[Sum[SeriesCoefficient[S, 2k] Zeta[2k], {k, 1, 200}], 70]], 50]
Block[{$MaxExtraPrecision = 1000}, Do[Print[N[1/Exp[Sum[(2^(2*n) - 1)*Zeta[2*n]^2/(n*Pi^(2*n)), {n, 1, m}]], 110]], {m, 250, 300}]] (* Vaclav Kotesovec, Sep 20 2014 *)
PROG
(PARI) exp(-sumpos(n=1, -log(cos(1/n)))) \\ warning: requires 2.6.2 or greater; Charles R Greathouse IV, Nov 04 2013
(PARI) T(n)=((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!
lm=lambertw(2*log(Pi/2)*10^default(realprecision))/2/log(Pi/2); exp(-sum(n=1, lm, T(n)*zeta(2*n))) \\ Charles R Greathouse IV, Nov 06 2013
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Fredrik Johansson, May 23 2006
EXTENSIONS
Corrected offset and extended by Robert G. Wilson v, Nov 03 2013
STATUS
approved