On partition of integers I came across an example in my textbook where it was asked to find the generating function for the number of integer solutions of: 
${2w+3x+5y+7z=n}$ where ${0\le w,  4\le x,y,  5\le z}$      
The proposed solution is:
$$\frac1{1-t^2}\cdot\frac{t^{12}}{1-t^3}\cdot\frac{t^{20}}{1-t^5}\cdot\frac{t^{35}}{1-t^7}$$
I do understand the generating functions for these partitions, I do not however grasp the reason why the numerator for instance when ${4\le x,y}$ is ${t^{12}}$ and ${t^{20 }}$ respectively. Or why it is ${t^{35}}$ when ${5\le z}$

Take $x$ as an example; it corresponds to the factor $\dfrac{t^{12}}{1-t^3}$. Now
$$\frac1{1-t^3}=\sum_{n\ge 0}t^{3n}=1+t^3+t^6+t^9+\ldots+t^{3k}+\ldots\;.\tag{1}$$
When you multiply the four series together, a typical term $t^{3k}$ of this series would contribute $3k$ to the total exponent of any term in the product in which it’s involved; that would correspond to $x=k$. However, you don’t want to allow $x$ to be $0,1,2$, or $3$, so you need to get rid of the terms $1,t^3,t^6$, and $t^9$: you want to start with $t^{12}=t^{3\cdot4}$. Multiplying $(1)$ by $t^{12}$ does exactly that: you get
$$\frac{t^{12}}{1-t^3}=t^{12}\sum_{n\ge 0}t^{3n}=\sum_{n\ge 0}t^{12+3n}=t^{12}+t^{15}+t^{18}+\ldots\;.\tag{1}$$
A similar explanation applies to the other numerators that aren’t $1$: $20=5\cdot 4$, and $35=7\cdot 5$, in each case giving you the minimum acceptable exponent.
