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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).
4
8, 96, 960, 161280, 2903040, 638668800, 49816166400, 83691159552000, 2845499424768000, 1946321606541312000, 408727537373675520000, 48662619743783485440000, 124089680346647887872000000, 174221911206693634572288000000, 70734095949917615636348928000000
OFFSET
1,1
COMMENTS
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
LINKS
FORMULA
A276592(n)/a(n) + A276594(n)/A276595(n) = A046988(n)/A002432(n).
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
EXAMPLE
From Seiichi Manyama, Sep 03 2018: (Start)
n | Pi^(2*n) | a(n)/A276592(n)
--+---------------+------------------------------------
1 | 9.8... | 8
2 | 97.4... | 96
3 | 961.3... | 960
4 | 9488.5... | 161280/17 = 9487.0...
5 | 93648.0... | 2903040/31 = 93646.4...
6 | 924269.1... | 638668800/691 = 924267.4...
7 | 9122171.1... | 49816166400/5461 = 9122169.2... (End)
MAPLE
seq(denom(sum(1/(2*k-1)^(2*n), k=1..infinity)/Pi^(2*n)), n=1..22);
MATHEMATICA
a[n_]:=Denominator[(1-2^(-2 n)) Zeta[2 n]] (* Steven Foster Clark, Mar 10 2023 *)
a[n_]:=Denominator[1/2 SeriesCoefficient[1/(E^x+1), {x, 0, 2 n-1}]] (* Steven Foster Clark, Mar 10 2023 *)
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 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Martin Renner, Sep 07 2016
STATUS
approved