What is $\sum_{k=1}^m\left[2^k\begin{pmatrix} n \\ k \end{pmatrix}\right]^2$?

Question

The equations $$ \sum_{k=1}^m\begin{pmatrix} n \\ k \end{pmatrix}^2 \quad \text{and} \quad \sum_{k=1}^m\left[2^k\begin{pmatrix} n \\ k \end{pmatrix}\right]^2$$ popped up in some of my calculations, and I was wondering, if there is an elegant solution to it. $\begin{pmatrix} n \\ k \end{pmatrix}$ is the binomial coefficient. The only thing, I found so far is that $$ \sum_{k=1}^n\begin{pmatrix} n \\ k \end{pmatrix}^2 = \begin{pmatrix} 2n \\ n \end{pmatrix} -1, $$ but in my case, $m<n$. Thank you for any leads!