From the Book Table of Integrals, Series and Products, on page 9, equation 0.235 says: $$ S_n = \sum_{k=1}^{\infty} \frac{1}{(4k^2 - 1)^n}, $$ with the first few values $$ S_1 = \frac{1}{2}, \quad S_2 = \frac{\pi^2 - 8}{16}, \quad S_3 = \frac{32 - 3\pi^2}{64}, \quad S_4 = \frac{\pi^4 + 30\pi^2 - 384}{768} $$
What is the general formula or recurrence relation?
If you want to better understand, look at $$S_n = \sum_{k=1}^{\infty} \frac{x^{2k}}{(4k^2 - 1)^n}$$ and you will see terms in $x$, $\tanh^{-1}(x)$ and Lerch phi functions.
Now, making $x=1$ lead to the specific values. For example $$S_{10}=$$ $$ \frac{-1486356480+68918850 \pi ^2+4729725 \pi ^4+270270 \pi ^6+8415 \pi ^8+62 \pi ^{10}}{2972712960}$$