Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $ I got this sum, in some work related to another question:
$$S_m=\sum_{k=1}^m  \frac{1}{k}{m \choose k} $$
Are there any known results about this (bounds, asymptotics)? 

A rather simple and gross bound is $$\sum_{k=1}^m\frac{1}{k}\binom{m}{k}\leq\sum_{k=1}^m\binom{m}{k}<\sum_{k=0}^m\binom{m}{k}=2^m$$
