Do similar expressions exist for odd zeta values other than $3$? Since Hjortnaes (and later Apéry), we know that
$$
\zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {k^{3}\binom {2k}{k}}}.
$$
I read somewhere that there might be a similar expression for $\zeta(5)$:
$$
\zeta(5)=\frac{a}{b} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {k^{5}\binom {2k}{k}}},
$$
where $a$ and $b$ are positive nonzero integers. I know that no such $a$ and $b$ have been found yet, otherwise Apéry's proof of the irrationality of $\zeta(3)$ could be extended to $\zeta(5)$. Are there any results suggesting that there exist similar sums for $\zeta(2n+1)$, $n>1$:
$$
\zeta(2n+1)=\frac{a}{b} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {k^{2n+1}\binom {2k}{k}}}?
$$
Conversely, are there any results that rule out the existence of such sums for certain odd positive integers?
Finally, what are the current known bounds on $a$ and $b$, if such sums can exist?
 A: The first terms of the continued fraction of 
$$ \frac{1}{\zeta(5)}\sum_{k\geq 1}\frac{(-1)^{k+1}}{k^5 \binom{2k}{k}} $$
are given by 
$$[0; 2, 10, 1, 1, 5, 1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 66, 4, 135, 3, 1, \
6, 1, 29,\ldots]$$
so if the previous number is $\frac{a}{b}\in\mathbb{Q}$, we have $\min(a,b)>3\cdot 10^{12}$.
In other terms, it looks pretty unlikely that the series $\sum_{k\geq 1}\frac{(-1)^{k+1}}{k^5 \binom{2k}{k}}$ provides a rational multiple of $\zeta(5)$.
The classical results
$$ \sum_{n\geq 1}\frac{1}{n^2\binom{2n}{n}}=\frac{1}{3}\zeta(2),\quad \sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\binom{2n}{n}}=\frac{2}{5}\zeta(3),\quad \sum_{n\geq 1}\frac{1}{n^4\binom{2n}{n}}=\frac{17}{36}\zeta(4)$$
(stating that in some cases the hypergeometric $\phantom{}_{p+1} F_p\left(1,1,1\ldots;\tfrac{3}{2},2,2,\ldots;\pm\tfrac{1}{4}\right)$ has a nice closed form in terms of the $\zeta$ function) can be seen as consequences of creative telescoping, or as consequences of symmetry relations for polylogarithms (see [1]). About $\frac{1}{k^5}$, creative telescoping produces spurious terms and the functional relations for $\text{Li}_4$ and $\text{Li}_5$ are pretty messy, so there are substantial obstructions in generalizing the classical approaches for producing a simple irrationality proof for $\zeta(5)$.
