Closed form of this bi-variate recurrent relation: $a_{i,j} = a_{i-1,j-1} + (r_i a_{i,j-1}) + a_{i+1,j-1}$

Question

How to find the closed form of this relation?

$\forall i,j \ge 1, a_{i,j} = a_{i-1,j-1} + (r_i a_{i,j-1}) + a_{i+1,j-1}$
where $\forall i \ge 0, a_{i,0}\in\{0,1\}$ and $r_i\in\{0,1\}$