This is an interesting problem. From my thinking on it, I don't think it's possible to write this only using one sigma. The reason I think so is that you have too many things to keep track of and not enough variables in your current explanation. The total sum you seek is defined by other sums, so you must either define the smaller sums with another sigma or find a nice pattern in the sums that allows you to express the current total in terms of N.
I thought of it like this:
and 1 < n < N.
We must find a way to relate the increasing value of "n" and the decreasing value of "N" in terms of one variable. I can't think of a way to do it. Maybe some of these thoughts will be useful or someone else can solve it.