This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A050971 #45 Feb 16 2025 08:32:41
%S A050971 1,2,8,24,384,240,46080,40320,2064384,725760,3715891200,159667200,
%T A050971 392398110720,12454041600,1428329123020800,20922789888000,
%U A050971 274239191619993600,711374856192000,1678343852714360832000
%N A050971 4*Denominator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} ((4k+1)^(-n)).
%C A050971 Reduced denominators of the Favard constants.
%H A050971 Theo Niessink, <a href="/A050971/b050971.txt">Table of n, a(n) for n = 1..200</a> (uploaded again by _Georg Fischer_, Feb 20 2019)
%H A050971 N. D. Elkies, <a href="https://arxiv.org/abs/math/0101168">On the sums Sum((4k+1)^(-n),k,-inf,+inf)</a>, arXiv:math/0101168 [math.CA], 2001-2003.
%H A050971 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 A050971 Maths StackExchange, <a href="https://math.stackexchange.com/questions/4112014/can-this-equation-be-written-in-terms-of-x">Can this equation be written in terms of x?</a>, Apr 22 2021.
%H A050971 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FavardConstants.html">Favard Constants.</a>
%F A050971 There is a simple formula in terms of Euler and Bernoulli numbers.
%e A050971 The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ...
%p A050971 S := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else
%p A050971 S(n, k - 1) + S(n - 1, n - k) fi end: EZ := n -> S(n, n)/(2^n*n!):
%p A050971 A050971 := n -> denom(EZ(n-1)): seq(A050971(n), n=1..19); # _Peter Luschny_, Aug 02 2017
%t A050971 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 *)
%t A050971 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 *)
%t A050971 Table[4/(2 Pi)^n LerchPhi[(-1)^n, n, 1/2], {n, 21}] // Denominator (* _Eric W. Weisstein_, Aug 02 2017 *)
%t A050971 Table[4/Pi^n If[Mod[n, 2] == 0, DirichletLambda, DirichletBeta][n], {n, 21}] // Denominator (* _Eric W. Weisstein_, Aug 02 2017 *)
%Y A050971 Cf. A068205, A050970 (numerators).
%K A050971 nonn,frac
%O A050971 1,2
%A A050971 _Eric W. Weisstein_
%E A050971 Entry revised by _N. J. A. Sloane_, Mar 24 2002