Consider the Random walk on the non-negative integers with transition probabilities $$ p_{0,1}=1,~~~p_{i,i+1}=1-r,~~~p_{i,i-1}=r,~~~i\geq 1. $$ Determine $p_{00}^{(n)}$
As far as I see it is $$ p_{00}^{(2n)}=\frac{1}{n+1}\binom{2n}{n}(1-r)^{n-1}r^n,~~~~~p_{00}^{(2n-1)}=0~~~\forall n\in\mathbb{N}. $$.
I would like to know if I am right.
If not, please give me some help.
With greetings