Let $c \in \mathbb{R}$. I would like to solve the following non-constant, two-indices recurrence relation on $u = (u_j^n)^{n=1, 2, \dots}_{j=0, 1, \dots}$: $$ u_j^0 = 0, \qquad \text{for} \quad j \geq 0. $$ $$ u_0^{1} = 1, \qquad u_{j}^{1} = 0, \quad \text{for} \quad j = 1, 2, \dots, $$ and for $n = 1, 2, \dots$ $$ u_0^{n+1} = \tfrac{1}{2}(1+c)u_0^{n} -c u_1^{n}, \qquad u_{j}^{n+1} = cu_{j-1}^n - cu_{j+1}^n + u_{j}^{n-1}, \quad \text{for} \quad j = 1, 2, \dots. $$ Is there any way to find the expression of $u_j^n$ (take for example $c=1/2$ to simplify)? The straightforward result is that $u_{n-1}^n = c^{n-1}$ and that $u_{n-2}^n = (1+c)c^{n-2}$.
Many thanks!