Consider the series
$$f(z)=\sum\_{n\ge1}\dfrac{z^n}{n^2\binom{2n}{n}}$$
which converges for $|z|\le4$.


One has $f(0)=0$, $f(1)=\pi^2/18$, $f(2)=\pi^2/8$, $f(3)=2\pi^2/9$, and $f(4)=\pi^2/2$.
Naive question: are there any other reasonably "explicit" evaluations (for instance for
$z=-1$, $-2$, $1/2$...) ? Note that the series for $z=1$ is related to Ap'ery's proof of the irrationality of $\zeta(2)$.



Special values of $\sum_{n\ge1}z^n/(n^2\binom{2n}{n})$
I interpret the request of the OP for an "explicit" evaluation of the series as a request for a "closed-form" expression, which exists (it seems to go back to Euler, [here](https://math.stackexchange.com/a/383181/87355) are several proofs):
$$f(z)=\sum\_{n\ge1}\dfrac{z^n}{n^2\binom{2n}{n}}=2 \arcsin^2\,(\tfrac{1}{2}\sqrt{z}),\;\;|z|\leq 4.$$


