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A340472 Numerators of an approximation to zeta(n)/Pi^n.

%I A340472 #37 Jan 12 2021 21:43:46
%S A340472 1,1,1,1,5,1,61,1,277,1,50521,691,540553,2,199360981,3617,3878302429,
%T A340472 43867,2404879675441,174611,14814847529501,155366,69348874393137901,
%U A340472 236364091,238685140977801337,1315862,4087072509293123892361,6785560294,13181680435827682794403,6892673020804
%N A340472 Numerators of an approximation to zeta(n)/Pi^n.
%H A340472 Melchor Viso Martinez, <a href="http://canal-arte.net/sites/default/files/ZetaSums_1.pdf">An expression for integer zeta approximation</a>
%F A340472 a(n) = numerator 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 A340472 a(2*n) = A046988(n).
%e A340472 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 A340472 Values are approximate for odd indices, exact for even indices:
%e A340472   zeta(1) ~       1/2             zeta(2) = Pi^2/6
%e A340472   zeta(3) ~    Pi^3/28            zeta(4) = Pi^4/90
%e A340472   zeta(5) ~  5*Pi^5/1488          zeta(6) = Pi^6/945
%e A340472   zeta(7) ~ 61*Pi^7/182880,       zeta(8) = Pi^8/9450
%e A340472   ...
%t A340472 a[k_] := Numerator[(1/(4 (1 - 2^-k) k!)
%t A340472       D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda],
%t A340472        k}]) /. {\[Lambda] -> 0}]
%o A340472 (PARI) a(n) = {my(t=tan(x/4 + O(x*x^n))); numerator(polcoef(x*(1 + t)/(1 - t), n)/((4-2^(2-n))))} \\ _Andrew Howroyd_, Jan 10 2021
%Y A340472 Cf. A046988, A340471 (denominators).
%K A340472 nonn,frac
%O A340472 1,5
%A A340472 _Melchor Viso Martinez_, Jan 08 2021