cp's OEIS Frontend

A050970 Numerator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} (4k+1)^(-n).

%I A050970 #46 Feb 16 2025 08:32:41
%S A050970 1,1,1,1,5,1,61,17,277,31,50521,691,540553,5461,199360981,929569,
%T A050970 3878302429,3202291,2404879675441,221930581,14814847529501,4722116521,
%U A050970 69348874393137901,56963745931,238685140977801337,14717667114151
%N A050970 Numerator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} (4k+1)^(-n).
%C A050970 Reduced numerators of Favard constants.
%H A050970 Vincenzo Librandi, <a href="/A050970/b050970.txt">Table of n, a(n) for n = 1..200</a>
%H A050970 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.
%H A050970 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 A050970 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
%H A050970 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FavardConstants.html">Favard Constants</a>
%F A050970 There is a simple formula in terms of Euler and Bernoulli numbers.
%F A050970 a(2n) = A046976(n), a(2n+1) = A089171(n+1) (conjectured).
%F A050970 Numerator of coefficients of expansion of (sec(x/2) + tan(x/2) + 1)/2 in powers of x. - _Sergei N. Gladkovskii_, Nov 11 2014
%e A050970 The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ...
%p A050970 S := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else
%p A050970 S(n, k - 1) + S(n - 1, n - k) fi end: EZ := n -> S(n, n)/(2^n * n!):
%p A050970 A050970 := n -> numer(EZ(n-1)): seq(A050970(n), n=1..26); # _Peter Luschny_, Aug 02 2017
%p A050970 # alternative
%p A050970 A050970 := proc(n)
%p A050970     if type(n,'even') then
%p A050970         (-1)^(n/2)*2^(n-2)/(n-1)!*euler(n-1,0) ;
%p A050970     else
%p A050970         (-1)^((n-1)/2)*2^(n-2)/(n-1)!*euler(n-1,1/2) ;
%p A050970     end if;
%p A050970     %/2^n ;
%p A050970     numer(%) ;
%p A050970 end proc:
%p A050970 seq(A050970(n),n=1..20) ; # _R. J. Mathar_, Jun 26 2024
%t A050970 s[n_] := Sum[(4*k + 1)^(-n), {k, -Infinity, Infinity}]; a[n_] := Numerator[FullSimplify[s[n]/Pi^n]]; a[1] = 1; Table[a[n], {n, 1, 26}] (* _Jean-François Alcover_, Oct 25 2012 *)
%t A050970 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] // Numerator, {n, 1, 26}] (* _Jean-François Alcover_, May 13 2013 *)
%t A050970 a[n_] := 4*Sum[((-1)^k/(2*k+1))^n, {k, 0, Infinity}] /. Pi -> 1 // Numerator; Table[a[n], {n, 1, 26}] (* _Jean-François Alcover_, Jun 20 2014 *)
%t A050970 Table[4/(2 Pi)^n LerchPhi[(-1)^n, n, 1/2], {n, 21}] // Numerator (* _Eric W. Weisstein_, Aug 02 2017 *)
%t A050970 Table[4/Pi^n If[Mod[n, 2] == 0, DirichletLambda, DirichletBeta][n], {n, 21}] // Numerator (* _Eric W. Weisstein_, Aug 02 2017 *)
%o A050970 (PARI) {a(n) = if( n<0, 0, numerator( polcoeff( 1 / (1 - tan(x/4 + x * O(x^n))), n)))}; /* _Michael Somos_, Nov 11 2014 */
%Y A050970 Denominators: A068205. See also A050971.
%Y A050970 Cf. A000828, A046982.
%K A050970 nonn,frac
%O A050970 1,5
%A A050970 _Eric W. Weisstein_
%E A050970 Entry revised by _N. J. A. Sloane_, Mar 24 2002