Number of integer solutions by generating functions method

I'm stuck in the middle of a problem and not sure where to go next. The original problem is:

Find the number of integer solutions to the equation

$$2x + 3y + 4z + w + s + t = n$$ with $$0 \le w \le 2$$ $$2 \le s \le 5$$ $$0 \le t \le 3$$

Now I was able to create my generating functions equations to get my overall equation to this:

$$G(x) = (\frac{1}{1-x^2})(\frac{1}{1-x^3}) (\frac{1}{1-x^4})(1+x+x^2)(x^2+x^3+x^4+x^5)(1+x+x^2+x^3)$$

Which with the help of the finite geometric series and some cancelling I was able to simplify down to:

$$G(x) = (\frac{1}{1-x^2})(\frac{x^2}{1}) (\frac{1}{(1-x)^3})(\frac{1-x^4}{1})$$

But now I'm stuck. Any help would be appreciated. Thanks.

Edit: With the extra step of simplification

$$G(x) = (\frac{x^2}{1}) (\frac{1}{(1-x)^3})(\frac{1+x^2}{1})$$

• Presumably $x,y,z$ are nonnegative integers? – Robert Israel Apr 21 '14 at 1:38
• Yes they all are – KwakKwak Apr 21 '14 at 1:51
• It would be better not to reuse $x$ between the equation and your generating function. Also if you use \left( and \right) in your $\LaTeX$ the parentheses expand to be large enough to enclose everything inside. – Ross Millikan Apr 21 '14 at 2:02

Hint: you can factor $1-x^2$ and $1-x^4$.