We have:
$$\frac{1-\sqrt{1-4x}}{2x} =\sum_{n= 0}^{\infty} \frac{1}{n+1}\binom{2n}{n}x^n $$
This is a well-known result about the generating function of Catalan numbers.
Now, I'm curious and really eager to know if we have such closed form for
$$\sum_{n= 0}^{\infty} \binom{2n}{n}^2x^n$$
Question: After some attempts, I failed all so I wonder if is there really a closed form?
A bit of context :So why this question may bear some interesting things? Because, this is a natural thing we may come up if we calculate the the expectation of number of times a two dimensional random walk returning to 0. That is, let $(S_n)$ be a simple symmetric random walks in $\mathbb{Z}^2$, $L_N = \sum_{k=0}^N \mathbb{1}_{S_k=(0,0)}$. Then
$$\mathbb{E}(L_{2N})=\sum_{n=0}^N \binom{2n}{n}^2\frac{1}{16^n}$$
Remark: $\mathbb{E}(L_{2N})= \Theta(\log N)$