Prove the identity $\sum\limits_{k=1}^{n+1} \sum\limits_{m=0}^{k-1} \binom{n+1}{k} \binom{n}{m} = 2^{2n}$

Question

How do I prove the following identity involving binomial coefficients?

$$\sum\limits_{k=1}^{n+1} \sum\limits_{m=0}^{k-1} \binom{n+1}{k} \binom{n}{m} = 2^{2n}$$

I understand that it can be proven with induction, however I would love to see combinatorial (intuitive) or algebraic solution.