Let $S = \{1,2,3,4,5,6\}$ and
$$P = \frac{1}{6}\begin{pmatrix} 3 & 3 & 0 & 0 & 0 & 0\\ 1 & 2 & 0 & 3 & 0 & 0\\ 0 & 1 & 4 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 5 & 0\\ 0 & 0 & 0 & 0 & 3 & 3\\ 0 & 0 & 0 & 0 & 1 & 5\\ \end{pmatrix}.$$
Determine $(E(s^{H_3}))^6_{i=1}$ (for $s \in (−1, 1)$).
My attempt: $H_3 = \inf\{n \geq 0: X_n = 3\}$ and $$E_i(s^{H_3})) = E(s^{H_3}\mid X_1 = i)) =\sum_{n<\infty} s^n P(H_3 = n).$$
So $$E_1(s^{H_3})) = sP_1(H_3 = 1) + s^2 P_1(H_3 = 2) + ... +s^6P_1(H_3 = 6).$$
I'm not sure how to calculate each $P_1(H_3 = n)$.