Binomial sum of $n$ terms in closed form Can we calculate the given combinatorial sum in closed form?
$$ \frac{\binom{2}{0}}{1}+\frac{\binom{4}{1}}{2}+\frac{\binom{8}{2}}{3}+\frac{\binom{16}{3}}{4}+\cdots+\frac{\binom{2^n}{n-1}}{n}$$

It is doubtful that a closed form exists for this sum. 
However, the terms grow very quickly. 
For $n$ large the sum is well approximated by its last term,
$$\sum_{k=1}^n \frac{1}{k}{2^k\choose k-1} \sim \frac{1}{n}{2^n\choose n-1}.$$ 
For $n=13$ the error due to this approximation is one part in a million. 
For $n=20$ the error is one part in ten billion. 
