A050971 4*Denominator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} ((4k+1)^(-n)).
1, 2, 8, 24, 384, 240, 46080, 40320, 2064384, 725760, 3715891200, 159667200, 392398110720, 12454041600, 1428329123020800, 20922789888000, 274239191619993600, 711374856192000, 1678343852714360832000
Offset: 1
Examples
The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ...
Links
- Theo Niessink, Table of n, a(n) for n = 1..200 (uploaded again by _Georg Fischer_, Feb 20 2019)
- 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.
- Maths StackExchange, Can this equation be written in terms of x?, Apr 22 2021.
- Eric Weisstein's World of Mathematics, Favard Constants.
Programs
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Maple
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!): A050971 := n -> denom(EZ(n-1)): seq(A050971(n), n=1..19); # Peter Luschny, Aug 02 2017
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Mathematica
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 *)
Formula
There is a simple formula in terms of Euler and Bernoulli numbers.
Extensions
Entry revised by N. J. A. Sloane, Mar 24 2002
Comments