OIM 2012 Problem 3

Let $n$ to be a positive integer. Given a set $\{ a_1, a_2, \ldots, a_n \}$ of integers, where $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$ $\forall i$ , we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to $0$ . If all of these sums have different remainders when divided by $2^n$ , we say that $\{ a_1, a_2, \ldots, a_n \}$ is $n$ -complete.

For each $n$ , find the number of $n$ -complete sets.