Count the Number of Houses at a Certain Distance I

Given n houses in a line with one extra edge between x and y, count for each k the number of ordered house pairs whose shortest path length equals k. The challenge is to avoid O(n^2) all-pairs checks by using the fact that distances are min(|i-j|, |i-x|+1+|j-y|, |i-y|+1+|j-x|) and aggregating counts with arithmetic in O(n).