When is the sum non-zero?

Question

Is there an easy way to show that the following expression is non-zero only for $l=1$?

$$\sum_{k = 0}^{l} \binom{l}{k} \binom{\frac{1}{2}\left(l + k - 1\right)}{l} \frac{1 + (-1)^{k+1}}{k+2}$$