%I #39 Apr 02 2023 00:42:43

%S 8,96,960,161280,2903040,638668800,49816166400,83691159552000,

%T 2845499424768000,1946321606541312000,408727537373675520000,

%U 48662619743783485440000,124089680346647887872000000,174221911206693634572288000000,70734095949917615636348928000000

%N Denominator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2*k-1)^(2*n).

%C A276592(n)/a(n) * Pi^(2*n) = Sum_{k>=1} 1/(2*k-1)^(2*n) > 1. So Pi^(2*n) > a(n)/A276592(n). - _Seiichi Manyama_, Sep 03 2018

%H Seiichi Manyama, <a href="/A276593/b276593.txt">Table of n, a(n) for n = 1..225</a>

%F A276592(n)/a(n) + A276594(n)/A276595(n) = A046988(n)/A002432(n).

%F A276592(n)/a(n) = (-1)^(n+1) * B_{2*n} * (2^(2*n) - 1) / (2 * (2*n)!), where B_n is the Bernoulli number. - _Seiichi Manyama_, Sep 03 2018

%e From _Seiichi Manyama_, Sep 03 2018: (Start)

%e n | Pi^(2*n) | a(n)/A276592(n)

%e --+---------------+------------------------------------

%e 1 | 9.8... | 8

%e 2 | 97.4... | 96

%e 3 | 961.3... | 960

%e 4 | 9488.5... | 161280/17 = 9487.0...

%e 5 | 93648.0... | 2903040/31 = 93646.4...

%e 6 | 924269.1... | 638668800/691 = 924267.4...

%e 7 | 9122171.1... | 49816166400/5461 = 9122169.2... (End)

%p seq(denom(sum(1/(2*k-1)^(2*n),k=1..infinity)/Pi^(2*n)),n=1..22);

%t a[n_]:=Denominator[(1-2^(-2 n)) Zeta[2 n]] (* _Steven Foster Clark_, Mar 10 2023 *)

%t a[n_]:=Denominator[1/2 SeriesCoefficient[1/(E^x+1),{x,0,2 n-1}]] (* _Steven Foster Clark_, Mar 10 2023 *)

%t a[n_]:=Denominator[1/2 Residue[Zeta[s] Gamma[s] (1-2^(1-s)) x^(-s),{s,1-2 n}]] (* _Steven Foster Clark_, Mar 11 2023 *)

%Y Cf. A002432, A046988, A276592, A276594, A276595.

%K nonn,frac

%O 1,1

%A _Martin Renner_, Sep 07 2016