A276592 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).

1, 1, 1, 17, 31, 691, 5461, 929569, 3202291, 221930581, 4722116521, 56963745931, 14717667114151, 2093660879252671, 86125672563201181, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 14129659550745551130667441, 16103843159579478297227731

Offset

1,4

Comments

Apart from signs, same as A089171 and A279370. - Peter Bala, Feb 07 2019

Links

Seiichi Manyama, Table of n, a(n) for n = 1..276

Siddharth Dwivedi, Vivek Kumar Singh, and Abhishek Roy, Semiclassical limit of topological Rényi entropy in 3d Chern-Simons theory, arXiv:2007.07033 [hep-th], 2020. See also J. of High Energy Physics (2020) Vol. 2020, Issue 12, Article 132.

Formula

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

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

Maple

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

Mathematica

a[n_]:=Numerator[Pi^(-2 n) (1-2^(-2 n)) Zeta[2 n]]  (* Steven Foster Clark, Mar 10 2023 *)
a[n_]:=Numerator[(-1)^n SeriesCoefficient[1/(E^x+1), {x, 0, 2 n-1}]] (* Steven Foster Clark, Mar 10 2023 *)
a[n_]:=Numerator[(-1)^n Residue[Zeta[s] Gamma[s] (1-2^(1-s)), {s, 1-2 n}]] (* Steven Foster Clark, Mar 11 2023 *)

Crossrefs

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

Sequence in context: A146462 A089171 A279370 * A002425 A275994 A046990

Adjacent sequences: A276589 A276590 A276591 * A276593 A276594 A276595

Keyword

nonn,frac

Author

Martin Renner, Sep 07 2016

Status

approved