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A050971
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4*Denominator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} ((4k+1)^(-n)).
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12
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1, 2, 8, 24, 384, 240, 46080, 40320, 2064384, 725760, 3715891200, 159667200, 392398110720, 12454041600, 1428329123020800, 20922789888000, 274239191619993600, 711374856192000, 1678343852714360832000
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OFFSET
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1,2
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COMMENTS
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Reduced denominators of the Favard constants.
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LINKS
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FORMULA
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There is a simple formula in terms of Euler and Bernoulli numbers.
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EXAMPLE
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The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ...
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MAPLE
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S := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else
S(n, k - 1) + S(n - 1, n - k) fi end: EZ := n -> S(n, n)/(2^n*n!):
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MATHEMATICA
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s[n_] := Sum[(4*k + 1)^(-n), {k, -Infinity, Infinity}]; a[n_] := 4*s[n]/Pi^n ; a[1] = 1; Table[a[n], {n, 1, 19}] // Denominator (* Jean-François Alcover, Nov 05 2012 *)
a[n_] := 4*Sum[((-1)^k/(2*k+1))^n, {k, 0, Infinity}] /. Pi -> 1 // Denominator; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 20 2014 *)
Table[4/(2 Pi)^n LerchPhi[(-1)^n, n, 1/2], {n, 21}] // Denominator (* Eric W. Weisstein, Aug 02 2017 *)
Table[4/Pi^n If[Mod[n, 2] == 0, DirichletLambda, DirichletBeta][n], {n, 21}] // Denominator (* Eric W. Weisstein, Aug 02 2017 *)
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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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EXTENSIONS
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STATUS
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approved
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