OFFSET
1,1
LINKS
Melchor Viso Martinez, An expression for integer zeta approximation
FORMULA
a(n) = denominator of lim_{x->0} of the n-th derivative of x*tan((Pi+x)/4)/((4-2^(2-n))*n!) with respect to x.
a(2*n) = A002432(n).
From Andrew Howroyd, Jan 10 2021: (Start)
a(n) = denominator of (1/(4-2^(2-n)))*[x^n] x*(1 + tan(x/4))/(1 - tan(x/4)).
a(n) = denominator( A000831(n-1)/((n-1)!*2^n*(2^n-1)) ). (End)
EXAMPLE
1/2, 1/6, 1/28, 1/90, 5/1488, 1/945, 61/182880, 1/9450, 277/8241408, 1/93555, 50521/14856307200, 691/638512875, ...
Values are approximate for odd indices, exact for even indices:
zeta(1) ~ 1/2 zeta(2) = Pi^2/6
zeta(3) ~ Pi^3/28 zeta(4) = Pi^4/90
zeta(5) ~ 5*Pi^5/1488 zeta(6) = Pi^6/945
zeta(7) ~ 61*Pi^7/182880, zeta(8) = Pi^8/9450
...
MATHEMATICA
a[k_] := Denominator[(1/(4 (1 - 2^-k) k!)
D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda],
k}]) /. {\[Lambda] -> 0}]
PROG
(PARI) a(n) = {my(t=tan(x/4 + O(x*x^n))); denominator(polcoef(x*(1 + t)/(1 - t), n)/((4-2^(2-n))))} \\ Andrew Howroyd, Jan 10 2021
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Melchor Viso Martinez, Jan 08 2021
STATUS
approved