Proof of a conjecture on the slit plane problem

Let ai,j(n) denote the number of walks in n steps from (0,0) to (i,j), with steps (±1,0) and (0,±1), never touching a point (−k,0) with k⩾0 after the starting point. Bousquet-Mélou and Schaeffer conjectured a closed form for the number a−i,i(2n) when i⩾1. In this paper, we prove their conjecture, and give a formula for a−i,i(2n) for i⩽−1.