For positive integer $n$ the following value of a hypergeometric function
$$_4F_3(n,n,n,2n,1+n,1+n,1+n,-1)$$
based on the first few terms looks like
$$ R_1(n) + R_2(n) \pi^2$$
where $R_{1,2}(n)$ are rational numbers. Is there a way to express $R_{1,2}(n)$ in a closed form using “simple” operations like power, ratio, factorial, etc?
Fiddling with Maple, I get: if $n$ is a positive real number, then $$ {{_4\mathrm F_3}(n,n,n,2\,n;\,n+1,n+1,n+1;\,-1)}={\frac {{n}^{2} \sqrt {\pi}\,\Gamma(n+1)\,\psi^{(1)}(n)}{{4}^{n}\,\Gamma \left( n +{\frac{1}{2}}\right)}. } $$ Here, $\psi^{(1)}$ is the trigamma function, $\psi^{(1)}(z) = \frac{d^2}{dz^2}\log(\Gamma(z))$.
It seems, for positive integer $n$, we have that $\psi^{(1)}(n)$ has the form $\pi^2/6 - \text{rational}$.
Francois pointed out the recurrence, so that $$ \psi^{(1)}(n) = \frac{\pi^2}{6}-\sum_{k=1}^{n-1}\frac{1}{k^2} $$
