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

4, 32, 1536, 184320, 8257536, 14863564800, 1569592442880, 5713316492083200, 1096956766479974400, 6713375410857443328000, 408173224980132554342400, 18857602994082124010618880000, 640578267860512766391484416000

Offset: 0

Author

N. J. A. Sloane, Feb 21 2000

Keywords

• nonn
• frac
• nice
• easy

Examples

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

References

• J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 384, Problem 15.
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Problem 37, beta(n).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Jan W. H. Swanepoel, A Short Simple Probabilistic Proof of a Well Known Identity and the Derivation of Related New Identities Involving the Bernoulli Numbers and the Euler Numbers, Integers (2025) Vol. 25, Art. No. A50. See p. 4.
• Eric Weisstein's World of Mathematics, Dirichlet Beta Function

Crossrefs

Cf. A046976.

Programs

Mathematica

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-François Alcover, Aug 19 2013 *)