Closed form solution for $\sum_{j=0}^{n} {n \choose j}^{2}x^{j}$?

Question

As the title suggests, I'm interested to see if there is a closed form solution to

\begin{equation}\label{q} \sum_{j=0}^{n} {n \choose j}^{2}x^{j}~. \end{equation}

This can be seen as a generalisation of

$$\sum_{j=0}^{n} {n \choose j}^{2} = {2n \choose n}~,$$

where $x=1$ in the title question.

Answer 1

For $n \in \mathbb{Z}^+$,

$$\sum\limits_{j=0}^n {n \choose j}^2 x^j = (1-x)^n P_n\left(\frac{1+x}{1-x}\right)$$

where $P_n( \cdot )$ is the Legendre polynomial.