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A340471 Denominators of an approximation to zeta(n)/Pi^n.

%I A340471 #52 Jan 12 2021 21:44:14
%S A340471 2,6,28,90,1488,945,182880,9450,8241408,93555,14856307200,638512875,
%T A340471 1569400842240,18243225,5713142135500800,325641566250,
%U A340471 1096948397364019200,38979295480125,6713362606110031872000,1531329465290625,408173030347971900211200,13447856940643125
%N A340471 Denominators of an approximation to zeta(n)/Pi^n.
%H A340471 Melchor Viso Martinez, <a href="http://canal-arte.net/sites/default/files/ZetaSums_1.pdf">An expression for integer zeta approximation</a>
%F A340471 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.
%F A340471 a(2*n) = A002432(n).
%F A340471 From _Andrew Howroyd_, Jan 10 2021: (Start)
%F A340471 a(n) = denominator of (1/(4-2^(2-n)))*[x^n] x*(1 + tan(x/4))/(1 - tan(x/4)).
%F A340471 a(n) = denominator( A000831(n-1)/((n-1)!*2^n*(2^n-1)) ). (End)
%e A340471 1/2, 1/6, 1/28, 1/90, 5/1488, 1/945, 61/182880, 1/9450, 277/8241408, 1/93555, 50521/14856307200, 691/638512875, ...
%e A340471 Values are approximate for odd indices, exact for even indices:
%e A340471   zeta(1) ~       1/2             zeta(2) = Pi^2/6
%e A340471   zeta(3) ~    Pi^3/28            zeta(4) = Pi^4/90
%e A340471   zeta(5) ~  5*Pi^5/1488          zeta(6) = Pi^6/945
%e A340471   zeta(7) ~ 61*Pi^7/182880,       zeta(8) = Pi^8/9450
%e A340471   ...
%t A340471 a[k_] := Denominator[(1/(4 (1 - 2^-k) k!)
%t A340471       D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda],
%t A340471        k}]) /. {\[Lambda] -> 0}]
%o A340471 (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
%Y A340471 Cf. A000831, A002432, A331839, A340472 (numerators).
%K A340471 nonn,frac
%O A340471 1,1
%A A340471 _Melchor Viso Martinez_, Jan 08 2021