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A340472
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Numerators of an approximation to zeta(n)/Pi^n.
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1
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1, 1, 1, 1, 5, 1, 61, 1, 277, 1, 50521, 691, 540553, 2, 199360981, 3617, 3878302429, 43867, 2404879675441, 174611, 14814847529501, 155366, 69348874393137901, 236364091, 238685140977801337, 1315862, 4087072509293123892361, 6785560294, 13181680435827682794403, 6892673020804
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OFFSET
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1,5
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
...
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MATHEMATICA
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a[k_] := Numerator[(1/(4 (1 - 2^-k) k!)
D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda],
k}]) /. {\[Lambda] -> 0}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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