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A068205 Denominator of S(n)/Pi^n, where S(n) = Sum((4k+1)^(-n),k,-inf,+inf).

%I A068205 #18 Jun 26 2024 10:20:36
%S A068205 4,8,32,96,1536,960,184320,161280,8257536,2903040,14863564800,
%T A068205 638668800,1569592442880,49816166400,5713316492083200,83691159552000,
%U A068205 1096956766479974400,2845499424768000,6713375410857443328000
%N A068205 Denominator of S(n)/Pi^n, where S(n) = Sum((4k+1)^(-n),k,-inf,+inf).
%H A068205 N. D. Elkies, <a href="http://arXiv.org/abs/math.CA/0101168">On the sums Sum((4k+1)^(-n),k,-inf,+inf)</a>, arXiv:math/0101168 [math.CA], 2001-2003.
%H A068205 N. D. Elkies, <a href="http://www.jstor.org/stable/3647742">On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n)</a>, Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
%H A068205 Z. K. Silagadze, <a href="https://arxiv.org/abs/1207.2055">Comment on the sums S(n) = sum(k=-inf..inf) 1/(4k+1)^n</a>, (2012) arXiv:1207.2055
%F A068205 There is a simple formula in terms of Euler and Bernoulli numbers.
%e A068205 The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ...
%p A068205 A068205 := proc(n)
%p A068205     if type(n,'even') then
%p A068205         (-1)^(n/2)*2^(n-2)/(n-1)!*euler(n-1,0) ;
%p A068205     else
%p A068205         (-1)^((n-1)/2)*2^(n-2)/(n-1)!*euler(n-1,1/2) ;
%p A068205     end if;
%p A068205     %/2^n ;
%p A068205     denom(%) ;
%p A068205 end proc:
%p A068205 seq(A068205(n),n=1..20) ; # _R. J. Mathar_, Jun 26 2024
%t A068205 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 *)
%t A068205 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 *)
%Y A068205 Numerators: A050970.
%K A068205 nonn,frac
%O A068205 1,1
%A A068205 _N. J. A. Sloane_, Mar 24 2002