If you have the generating function: $$(1+x+x^2+\dotso)$$ is the same as $$\frac{1}{(1-x)}$$
If you have the generating function (evens):
$$(1+x^2+x^4+x^6+\dotso)$$
is the same as
$$\frac{1}{(1-x^2)}$$
However, what happens when you have the generating function for odds?
i.e. $(x+x^3+x^5+x^7+\dotso)$
What does this "convert" to?
Since $$x+x^3+x^5... = x(1+x^2+x^5+...)$$ and $$1+x^2+x^5+...=\frac{1}{1-x^2} \ ,$$ we have that $$x+x^3+x^5... = x\left(\frac{1}{1-x^2}\right) =\frac{x}{1-x^2} \ .$$