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

4, 8, 32, 96, 1536, 960, 184320, 161280, 8257536, 2903040, 14863564800, 638668800, 1569592442880, 49816166400, 5713316492083200, 83691159552000, 1096956766479974400, 2845499424768000, 6713375410857443328000

Offset: 1

Author

N. J. A. Sloane, Mar 24 2002

Keywords

• nonn
• frac

Examples

			The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ...
		

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.
• Z. K. Silagadze, Comment on the sums S(n) = sum(k=-inf..inf) 1/(4k+1)^n, (2012) arXiv:1207.2055

Crossrefs

Numerators: A050970.

Programs

Maple

A068205 := proc(n)
    if type(n,'even') then
        (-1)^(n/2)*2^(n-2)/(n-1)!*euler(n-1,0) ;
    else
        (-1)^((n-1)/2)*2^(n-2)/(n-1)!*euler(n-1,1/2) ;
    end if;
    %/2^n ;
    denom(%) ;
end proc:
seq(A068205(n),n=1..20) ; # R. J. Mathar, Jun 26 2024

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 *)

Formula

There is a simple formula in terms of Euler and Bernoulli numbers.