APMO 2017 Problem 3

Let $A(n)$ denote the number of sequences $a_{1} \geq a_{2} \geq \ldots \geq a_{k}$ of positive integers for which $a_{1}+\cdots+a_{k}=n$ and each $a_{i}+1$ is a power of two $(i=1,2, \ldots, k)$. Let $B(n)$ denote the number of sequences $b_{1} \geq b_{2} \geq \ldots \geq b_{m}$ of positive integers for which $b_{1}+\cdots+b_{m}=n$ and each inequality $b_{j} \geq 2 b_{j+1}$ holds $(j=1,2, \ldots, m-1)$.

Prove that $A(n)=B(n)$ for every positive integer $n$.