A340471 Denominators of an approximation to zeta(n)/Pi^n.
2, 6, 28, 90, 1488, 945, 182880, 9450, 8241408, 93555, 14856307200, 638512875, 1569400842240, 18243225, 5713142135500800, 325641566250, 1096948397364019200, 38979295480125, 6713362606110031872000, 1531329465290625, 408173030347971900211200, 13447856940643125
Offset: 1
Examples
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 ...
Links
- Melchor Viso Martinez, An expression for integer zeta approximation
Programs
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Mathematica
a[k_] := Denominator[(1/(4 (1 - 2^-k) k!) D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda], k}]) /. {\[Lambda] -> 0}] -
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
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)