 Solution

Expand the analytic factor around zero by the binomial series:Every coefficient is nonnegative. Adding $2/3$ to the constant coefficient gives probabilities $p_j$ with sum $g(1)=2/3+1/3=1$. Define $\mathrm{Pr}(T=j)=p_j$ for $j\ge 0$. This is a nonnegative integer-valued random variable and its probability generating function is exactly $g$. The power series has radius $14/5>1$, so the evaluations and derivatives used below are finite.