For \{1,2,3, \ldots, n\} and each of its nonempty subsets a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for \{1,2,4,6,9\} is 9-6+4-2+1=6 and for \{5\} it is simply 5.) Find the sum of all such alternating sums for n=7.