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

1, 1, 1, 1, 5, 1, 61, 17, 277, 31, 50521, 691, 540553, 5461, 199360981, 929569, 3878302429, 3202291, 2404879675441, 221930581, 14814847529501, 4722116521, 69348874393137901, 56963745931, 238685140977801337, 14717667114151

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• frac

Comments

Reduced numerators of Favard constants.

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001.
• 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
• Eric Weisstein's World of Mathematics, Favard Constants

Crossrefs

Denominators: A068205. See also A050971.

Cf. A000828, A046982.

Programs

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!):
A050970 := n -> numer(EZ(n-1)): seq(A050970(n), n=1..26); # Peter Luschny, Aug 02 2017
# alternative
A050970 := 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 ;
    numer(%) ;
end proc:
seq(A050970(n),n=1..20) ; # R. J. Mathar, Jun 26 2024

Mathematica

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 *)
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 *)
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 *)
Table[4/(2 Pi)^n LerchPhi[(-1)^n, n, 1/2], {n, 21}] // Numerator (* Eric W. Weisstein, Aug 02 2017 *)
Table[4/Pi^n If[Mod[n, 2] == 0, DirichletLambda, DirichletBeta][n], {n, 21}] // Numerator (* Eric W. Weisstein, Aug 02 2017 *)

PARI

{a(n) = if( n<0, 0, numerator( polcoeff( 1 / (1 - tan(x/4 + x * O(x^n))), n)))}; /* Michael Somos, Nov 11 2014 */

Formula

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

a(2n) = A046976(n), a(2n+1) = A089171(n+1) (conjectured).

Numerator of coefficients of expansion of (sec(x/2) + tan(x/2) + 1)/2 in powers of x. - Sergei N. Gladkovskii, Nov 11 2014

Extensions

Entry revised by N. J. A. Sloane, Mar 24 2002