How to find the generating function for $U_n$ (the number of the solutions) for $2x+3y+5z=n,$ for $x,y\ge0, 10\ge z\ge0$?

Question

I need to find the generating function but I'm not sure how to do that. I understand that it is somehow related to:

$$f(t)= \frac 1{(1−t^2)(1−t^3)(1−t^5)}$$

but I don't know how to get there. I remember from the lectures that it is supposed to be somewhat like:

$$ \sum_{i=0}^n (x^2)^i \sum_{i=0}^n (x^3)^i \sum_{i=0}^{10} (x^5)^i $$

but I am not sure how to continue from here (if it is even correct). How do I find the generating function?

Answer 1

The $f(t)$ you show is close to correct. The first two terms are correct. The third needs to be adjusted to account for the upper bound of $z$. You want to contribute the following terms:

$$1+t^5+t^{2\cdot 5} + \cdots + t^{10\cdot 5} = \dfrac{1-t^{55}}{1-t^5}$$

so you can write

$$f(t) = \dfrac{1-t^{55}}{(1-t^2)(1-t^3)(1-t^5)}$$