if $\sum_0^\infty a_n x^n = (\sum_0^\infty x^n )(\sum_0^\infty x^{2n})$ what is $a_n$? if $$\sum_0^\infty a_n x^n = (\sum_0^\infty x^n )(\sum_0^\infty x^{2n})$$
what is $a_n$?
Here is my approach 
let $b_n= 1$ and $c_n= x^n$
Then by forming / relating to cauchy product we can conlude that the product is equal to : 
$$\sum_0^\infty a_n x^n$$
where $a_n = \sum_{k=0}^{n} b_k c_{n-k} = \sum x^{n-k}= x^n+x^{n-1}+...+x^0 = \frac{1-x^{n+1}}{1-x}$
What do you think about my approach? I feel like there is something wrong with it. If not - could you provide a different approach? By the way depending on the way I solve - I keep finding different answers. I also ended up with $a_n=\frac{3+2n+(-1)^n}{4}$ without using cauchy product.

Using the geometric summation formula $\displaystyle\sum x^n=\dfrac1{1-x}$,
$$A=\frac1{1-x}\frac1{1-x^2}=\frac1{4(1+x)}+\frac1{4(1-x)}+\frac1{2(1-x)^2}\\
=\frac1{4(1+x)}+\frac1{4(1-x)}+\frac12\frac d{dx}\frac x{1-x}$$ so that
$$a_n=\frac14(-1)^n+\frac14+\frac12(n+1).$$
This is a generating function approach.
