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%I A276593 #39 Apr 02 2023 00:42:43
%S A276593 8,96,960,161280,2903040,638668800,49816166400,83691159552000,
%T A276593 2845499424768000,1946321606541312000,408727537373675520000,
%U A276593 48662619743783485440000,124089680346647887872000000,174221911206693634572288000000,70734095949917615636348928000000
%N A276593 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 A276593 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 A276593 Seiichi Manyama, <a href="/A276593/b276593.txt">Table of n, a(n) for n = 1..225</a>
%F A276593 A276592(n)/a(n) + A276594(n)/A276595(n) = A046988(n)/A002432(n).
%F A276593 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 A276593 From _Seiichi Manyama_, Sep 03 2018: (Start)
%e A276593 n |    Pi^(2*n)   |   a(n)/A276592(n)
%e A276593 --+---------------+------------------------------------
%e A276593 1 |        9.8... |           8
%e A276593 2 |       97.4... |          96
%e A276593 3 |      961.3... |         960
%e A276593 4 |     9488.5... |      161280/17     =     9487.0...
%e A276593 5 |    93648.0... |     2903040/31     =    93646.4...
%e A276593 6 |   924269.1... |   638668800/691    =   924267.4...
%e A276593 7 |  9122171.1... | 49816166400/5461   =  9122169.2... (End)
%p A276593 seq(denom(sum(1/(2*k-1)^(2*n),k=1..infinity)/Pi^(2*n)),n=1..22);
%t A276593 a[n_]:=Denominator[(1-2^(-2 n)) Zeta[2 n]] (* _Steven Foster Clark_, Mar 10 2023 *)
%t A276593 a[n_]:=Denominator[1/2 SeriesCoefficient[1/(E^x+1),{x,0,2 n-1}]] (* _Steven Foster Clark_, Mar 10 2023 *)
%t A276593 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 A276593 Cf. A002432, A046988, A276592, A276594, A276595.
%K A276593 nonn,frac
%O A276593 1,1
%A A276593 _Martin Renner_, Sep 07 2016

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