A276592 - OEIS

%I #35 Apr 01 2023 23:04:15

%S 1,1,1,17,31,691,5461,929569,3202291,221930581,4722116521,56963745931,

%T 14717667114151,2093660879252671,86125672563201181,

%U 129848163681107301953,868320396104950823611,209390615747646519456961,14129659550745551130667441,16103843159579478297227731

%N Numerator 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 Apart from signs, same as A089171 and A279370. - _Peter Bala_, Feb 07 2019

%H Seiichi Manyama, <a href="/A276592/b276592.txt">Table of n, a(n) for n = 1..276</a>

%H Siddharth Dwivedi, Vivek Kumar Singh, and Abhishek Roy, <a href="https://arxiv.org/abs/2007.07033">Semiclassical limit of topological Rényi entropy in 3d Chern-Simons theory</a>, arXiv:2007.07033 [hep-th], 2020. See also <a href="https://doi.org/10.1007/JHEP12(2020)132">J. of High Energy Physics</a> (2020) Vol. 2020, Issue 12, Article 132.

%F a(n)/A276593(n) + A276594(n)/A276595(n) = A046988(n)/A002432(n).

%F a(n)/A276593(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

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

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

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

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

%Y Cf. A002432, A046988, A276593, A276594, A276595, A089171, A279370.

%K nonn,frac

%O 1,4

%A _Martin Renner_, Sep 07 2016