Solve this:

Let p(n) denote the number of distinct partitions of a positive integer n.

For example:

p(4) equals 5 because the partitions of 4 are: 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.
Now consider the following sum:

S(N) = sum from k=1 to N of [(-1)^(k+1) * k * p(N-k)]

Find the closed-form expression for S(N) in terms of N.

Prove that S(N) is always divisible by N for all positive integers N greater than or equal to 1.

Generalize the problem for any positive integer M, and determine if the sum S_M(N) = sum from k=1 to N of [(-1)^(k+1) * k^M * p(N-k)] is divisible by a function of N and M. Prove your result.