NZ-NZMO 2020 Problem 5

A sequence of $A s$ and $B s$ is called antipalindromic if writing it backwards, then turning all the $A s$ into $B s$ and vice versa, produces the original sequence. For example $A B B A A B$ is antipalindromic. For any sequence of $A s$ and $B s$ we define the cost of the sequence to be the product of the positions of the $A s$ . For example, the string $A B B A A B$ has cost $1\cdot 4\cdot 5 = 20$ . Find the sum of the costs of all antipalindromic sequences of length 2020.