I would like to evaluate the asymptotic value of following sum:

$\frac{1}{2^N}\sum_{n=0}^{N} \begin{pmatrix}N\\ n \end{pmatrix} \text{log}_{2} \begin{pmatrix}N\\ n \end{pmatrix}$. This is related to the computation of the Shannon entropy. Any help would be greatly appreciated!