Random walk on non-negative integers - Transcience Recurrence etc

Question

Good evening. I have a problem concerning the random walk on non-negative integers. Suppose that $ p_{0,1}=1 , p_{i,i+1}=1-p , p_{i,i-1}=p $ . I would like to know for which value of $ p $ this random walk is transient/recurrent. I tried to apply the classical theorem about calculating the probabilities $ p_{0,0}^{(2n)} $ and then try to tell if the series over $ n $ converges, but it seems really hard to find an explicit formula about these n-th transition probabilities. Does anybody have any idea about this?