Define a discrete-time markov chain with state space $I=\{0, 1, ..., d\}$ and transition probabilities given by:
$$p_{i, i+1} = \left(\frac{d-i}{d}\right)^2$$ $$p_{i, i} = \frac{2i(d-i)}{d^2}$$ $$p_{i, i-1} = \left(\frac{i}{d}\right)^2$$ $$p_{i,j} = 0 \quad , \text{ otherwise}.$$
How can one find its stationary distribution $\Pi = [\pi_0, ..., \pi_d]$?
I managed to calculate $\Pi$ for some small examples with $d=1,2,3$ and also could find a recurrence relation using the detailed-balance equation:
$$ \pi_i p_{i,j} = \pi_j p_{j,i} $$ concluding that:
$$ \pi_{i+1} = \left(\frac{d-i}{i+1}\right)^2\pi_i \quad, \quad i > 0$$
But I don't know how to find the answer for the general case.
Thanks in advance.