Consider a simple random walk in one dimension with reflective boundaries at $n=1$ and $n=N$. We can express it via the master equation: \begin{equation} P(n,t) = \frac{1}{2}P(n-1,t-1) + \frac{1}{2}P(n+1,t-1) \quad \text{if } n\neq 1 \;{\rm and} \;n\neq N, \end{equation} With the following boundary conditions: \begin{align} P(0,t) =P(N+1,t) = 0. \end{align}

I would like to know the exact expression of $P(n,t)$. I thought this would be simple to find but I can't find a reference with a simple expression for it... Am I missing something?