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A053005 Denominator of beta(2n+1)/Pi^(2n+1), where beta(m) = Sum_{k=0..inf} (-1)^k/(2k+1)^m. 2

%I

%S 4,32,1536,184320,8257536,14863564800,1569592442880,5713316492083200,

%T 1096956766479974400,6713375410857443328000,408173224980132554342400,

%U 18857602994082124010618880000,640578267860512766391484416000

%N Denominator of beta(2n+1)/Pi^(2n+1), where beta(m) = Sum_{k=0..inf} (-1)^k/(2k+1)^m.

%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 384, Problem 15.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Problem 37, beta(n).

%H T. D. Noe, <a href="/A053005/b053005.txt">Table of n, a(n) for n = 0..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>

%e beta(5) = 5*Pi^5/1536 so a(2)=1536.

%t beta[1] = Pi/4; beta[m_] := (Zeta[m, 1/4] - Zeta[m, 3/4])/4^m; a[n_, p_] := a[n, p] = beta[2*n+1]/Pi^(2*n+1) // N[#, p]& // Rationalize[#, 0]& // Denominator; a[n_] := Module[{p = 16}, a[n, p]; p = 2*p; While[a[n, p] != a[n, p/2], p = 2*p]; a[n, p]]; Table[a[n], {n, 0, 13}] (* _Jean-Fran├žois Alcover_, Aug 19 2013 *)

%Y Cf. A046976.

%K nonn,frac,nice,easy

%O 0,1

%A _N. J. A. Sloane_, Feb 21 2000

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Last modified January 29 11:47 EST 2023. Contains 359922 sequences. (Running on oeis4.)