How do you find the generating function for the number of integer solutions of this equation?

Question

I am halfway stuck on this problem, can someone explain to me how to continue from where I am stuck?

Question:3a + 5b +7c = n, where a,b,c >= 0.

So I first turned it into this

$$( 1 + x^3 + x^6...)(1 + x^5 +...)(1 + x^7+...)$$

How do I continue after this?

Answer 1

You have a product of geometric series. The first factor is $1/(1 - x^3)$, the second factor is $1/(1 - x^5)$, and the third factor is $1/(1 - x^7)$. So the result is

$$\frac{1}{(1 - x^3)(1 - x^5)(1 - x^7)}.$$