$\sum_{k=1}^{n} \frac{2^k-1}{k}$

$=\sum_{k=1}^{n} \frac{1}{k}(\sum_{j=1}^{k} \binom{k}{j})$

$=\sum_{j=1}^{n} \sum_{k=j}^{n} \binom{k}{j}\frac{1}{k}$

$=\sum_{j=1}^{n} \frac{1}{j}(\sum_{k=j}^{n} \binom{k-1}{j-1})$

$=\sum_{j=1}^{n} \frac{1}{j} \binom{n}{j}$