I've got this problem here:
Show that (1 - x - x^2 - x^3 - x^4 - x^5 - x^6)^-1 is the generating function for the number of ways a sum of r can occur if a die is rolled any number of times.
It's a homework problem, but it's past due, so I am just trying to figure it out because I think I may have missed something important that I need to solve this.
I was able to solve this with recurrence relations:
C0 = 1 (there is 1 way to roll a sum of 0), and CN = CN-1 + CN-2 + CN-3 + CN-4 + CN-5 + CN-6.
I used that to get h(x) - x - 2x^2 - 4x^3 - 8x^4 - 16x^5 - 32x^6 = x (h(x) - x - 2x^2 - 4x^3 - 8x^4 - 16x^5) + x^2 (h(x) - x - 2x^2 - 4x^3 - 8x^4) + x^3 (h(x) - x - 2x^2 - 4x^3) + x^4 (h(x) - x - 2x^2) + x^5 (h(x) - x) + x^6 (h(x)), where g(x) = h(x) + 1, because h(x) does not include the 1 * x^0 term. This works out fine.
But there should be some way to verify this without recurrence relations. I'm not sure how to convert the generating function to a sum, or how to reconstruct (build) the generating function, but there should be some way to do either of those.
Hopefully someone here can give some advice.