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Numerators of an approximation to zeta(n)/Pi^n.
1

%I #37 Jan 12 2021 21:43:46

%S 1,1,1,1,5,1,61,1,277,1,50521,691,540553,2,199360981,3617,3878302429,

%T 43867,2404879675441,174611,14814847529501,155366,69348874393137901,

%U 236364091,238685140977801337,1315862,4087072509293123892361,6785560294,13181680435827682794403,6892673020804

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

%H Melchor Viso Martinez, <a href="http://canal-arte.net/sites/default/files/ZetaSums_1.pdf">An expression for integer zeta approximation</a>

%F 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 a(2*n) = A046988(n).

%e 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 Values are approximate for odd indices, exact for even indices:

%e zeta(1) ~ 1/2 zeta(2) = Pi^2/6

%e zeta(3) ~ Pi^3/28 zeta(4) = Pi^4/90

%e zeta(5) ~ 5*Pi^5/1488 zeta(6) = Pi^6/945

%e zeta(7) ~ 61*Pi^7/182880, zeta(8) = Pi^8/9450

%e ...

%t a[k_] := Numerator[(1/(4 (1 - 2^-k) k!)

%t D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda],

%t k}]) /. {\[Lambda] -> 0}]

%o (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 Cf. A046988, A340471 (denominators).

%K nonn,frac

%O 1,5

%A _Melchor Viso Martinez_, Jan 08 2021