There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. The number is defined precisely below. To $60$ decimals it is \begin{align} v &= 2.4953225371\ 3460865878\ 0441528421\ 4735785377\ 0845907036\ 757056931 \\ \frac{1}{v} &= 0.4007497969\ 1735762467\ 103472853\ 2244486515\ 1108820032\ 4884686488 \end{align}
Question. What number is that?
A recent post by Tito Piezas III proposes values of continued fractions arising from certain three-term recurrences. It reminded me of this, which I did around 2014. Following Apéry's proofs of irrationality of $\zeta(2), \zeta(3)$, I worked out many similar examples. There were many continued fractions. In some cases (a) conjectured values were clear; in some cases (b) the Liouville proof that the value is irrational worked. But cases (which would be most interesting) where both (a) and (b) hold were very few. This is one of the (b) cases.
Continued fraction \begin{align} v &= \frac{5}{2} +{\Large{K}}_{n=1}^\infty\frac{-n^2(n-\frac12)^2}{\;34n^2+17n+\frac52\;} \\ 2v &= 5 + {\Large{K}}_{n=1}^\infty\frac{-n^2(2n-1)^2}{\;68n^2+34n+5\;} = 5 - \frac{1}{\displaystyle 107 -\frac{36}{\displaystyle345 - \frac{225}{\displaystyle719 - \ddots}}} \end{align}
Recurrences
The "pre-Apery" sequence $(a_n)$ is defined by the recurrence
$$
0 = (n+1)^2 a_{n+1}-\left(34n\left(n+\frac12\right)
+\frac52\right)a_n+\left(n-\frac12\right)^2a_{n-1}
\tag1$$
with initial values $a_0 = 1, a_1 = \frac52$.
I called it that because of the generating function equation
$$
\left(\sum_{n=0}^\infty a_n x^n\right)^2 = \sum_{n=0}^\infty A_n x^n
$$
where $(A_n)$ is the usual Apéry sequence. [Note $(4^n a_n)
=(1,10,534,40900,\cdots)$ is an integer sequence not listed in OEIS.]
A "complementary sequence" $(b_n)$ satisfies the same recurrence $(1)$ with initial values $b_0 = 0, b_1 = 1$.
\begin{align}
(a_n) &= \left(1, \frac52, \frac{267}{8}, \frac{10225}{16},
\frac{1836275}{128}, \frac{90191115}{256}, \frac{9386644575}{1024}, \frac{508540579305}{2048},\cdots\right),
\\
(b_n) &= \left(0, 1, \frac{107}{8},
\frac{12293}{48}, \frac{4415321}{768}, \frac{1807203551}{12800},
\frac{22570175443}{6144}, \frac{1497911874023249}{15052800},\cdots\right) .
\end{align}
Our value is:
$$
v = \lim_{n\to\infty}\frac{a_n}{b_n} .
$$
Rapidly converging series
$$
\frac{1}{v} = \sum_{k=0}^\infty
\frac{(2k)!^2}{2^{4k} k!^2 (k+1)!^2 a_k a_{k+1}}
$$
