Property of sum $\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}$

Question

Is it true that for all $n\in\mathbb{N}$, \begin{align}f(n)=\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}\end{align} is always rational. I have calculated via Mathematica, which says \begin{align}f(0)=\frac{1}{24},f(1)=\frac{31}{504},f(2)=\frac{511}{264},f(3)=\frac{8191}{24}\end{align} But I couldn't find the pattern or formula behind these numbers, Thanks for your help!

Answer 1

This series appears in Apostol's book "Modular Functions and Dirichlet Series in Number Theory" p.25 according to (8) from MathWorld with the result (if your series starts with $k=0$) : $$f(n)=\frac {2^{4n+1}-1}{8n+4}\,B_{4n+2}$$ with $B_n$ a Bernoulli number.

UPDATE: Apostol's book may be consulted here and the theorem $13.17$ is the proof of the classical relation between $\zeta(2n)$ and $B_{2n}$.