Let $c(n)$ denote the number of partitions of $n$ into non-negative powers of $2.$ (Thus $c(5)=4$ since $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1).$ (a). Prove that $1+\sum\limits_{n=1}^{\infty} c(n) q^n=\prod\limits_{n=0}^{\infty} (1-q^{2^n})^{-1}.$ (b). Use part $(a)$ to show that $c(2n+1)=c(2n)$ and $c(2n)=c(2n-1)+c(n).$
I tried to follow the same argument as Apostal(Intro. to Analytic Number Theory) done in his proof in Theorem 14.2. How can I distinguish the coefficient I get by expanding out each series as the one I want ?