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 A068205 Denominator of S(n)/Pi^n, where S(n) = Sum((4k+1)^(-n),k,-inf,+inf). 3
 4, 8, 32, 96, 1536, 960, 184320, 161280, 8257536, 2903040, 14863564800, 638668800, 1569592442880, 49816166400, 5713316492083200, 83691159552000, 1096956766479974400, 2845499424768000, 6713375410857443328000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001-2003. N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573. FORMULA There is a simple formula in terms of Euler and Bernoulli numbers. EXAMPLE The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ... MATHEMATICA s[n_?EvenQ] := (-1)^(n/2-1)*(2^n-1)*BernoulliB[n]/(2*n!); s[n_?OddQ] := (-1)^((n-1)/2)*2^(-n-1)*EulerE[n-1]/(n-1)!; Table[s[n] // Denominator, {n, 1, 19}] (* Jean-François Alcover, May 13 2013 *) a[n_] := 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 *) CROSSREFS Numerators: A050970. Sequence in context: A086344 A209084 A254216 * A241684 A254878 A247473 Adjacent sequences:  A068202 A068203 A068204 * A068206 A068207 A068208 KEYWORD nonn,frac AUTHOR N. J. A. Sloane, Mar 24 2002 STATUS approved

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