I try to find the probability $p_{n, k}$of the fact that simple symmetric visits the origin exactly $k$ times on the segment $[0, 2n]$, where $n\ge k$.
I tried to solve recurrent formula: $$p_{n, k} = \sum_{i=1}^{n-k}f_i\cdot p_{n-i, k-1},$$ where $f_i$ is the probability of the fact that the first return get in the point $2i$. But I can't simplify it. As another try I wrote the formula of inclusion-exclusion with an events when random walk visits the origin at the fixed point. But I can't figure it out. May be another simple way exists?