RO-TST 2012 Problem 18

Let $n$ be a positive integer. Find the value of the following sum $$\sum_{(n)}\sum_{k=1}^n {e_k2^{e_1+\cdots+e_k-2k-n}},$$ where $e_k\in\{0,1\}$ for $1\leq k \leq n$ , and the sum $\sum_{(n)}$ is taken over all $2^n$ possible choices of $e_1,\ldots ,e_n$ .