I would like to know how to prove the following identity.
Let m be a positive integer. Then 

$$\sum\_{i=0}^{m+1} (-1)^i \Bigl\lfloor \frac{i}{2} \Bigr\rfloor {2m+1 \choose m+i}=2^{2m-2}$$

Here $\Bigl\lfloor \frac{i}{2} \Bigr\rfloor$ denotes the floor function.

A computational proof is welcome. On the other hand, is it possible to give a combinatorial proof? For example, we interpret the left-hand side as coefficients of a variable in a polynomial.