The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A050970 Numerator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} (4k+1)^(-n). 15
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Reduced numerators of Favard constants. 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. Eric Weisstein's World of Mathematics, Favard Constants 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 EXAMPLE The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ... 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 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 *) PROG (PARI) {a(n) = if( n<0, 0, numerator( polcoeff( 1 / (1 - tan(x/4 + x * O(x^n))), n)))}; /* Michael Somos, Nov 11 2014 */ CROSSREFS Denominators: A068205. See also A050971. Cf. A000828, A046982. Sequence in context: A144268 A013988 A246006 * A138548 A220422 A251596 Adjacent sequences:  A050967 A050968 A050969 * A050971 A050972 A050973 KEYWORD nonn,frac AUTHOR EXTENSIONS Entry revised by N. J. A. Sloane, Mar 24 2002 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 6 13:57 EDT 2020. Contains 333276 sequences. (Running on oeis4.)