APMO 2017 - Asian Pacific Mathematics Olympiad Question 3

Let $A(n)$ be the number of sequences $a_1\ge\cdots\ge a_k$ of positive integers with sum $n$ such that every $a_i+1$ is a power of 2. Let $B(n)$ be the number of sequences $b_1\ge\cdots\ge b_m$ of positive integers with sum $n$ such that $b_j\ge2b_{j+1}$ for $1\le j\lt m$. Prove $A(n)=B(n)$ for every positive integer $n$.

Proposed by the Senior Problems Committee of the Australian Mathematical Olympiad Committee.