I want to calculate the probability of transition in n step from state 0 to state 0 ($p_{00}^{(n)}$) in below Markov-Chain :

if self loop in state 0 doesn't exist, probability computed with Catalan Number. but in this form, how can calculate?
I want to calculate the probability of transition in n step from state 0 to state 0 ($p_{00}^{(n)}$) in below Markov-Chain :

if self loop in state 0 doesn't exist, probability computed with Catalan Number. but in this form, how can calculate?
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