Suppose that $Y$ is a random variable such that $E(Y^k) = \frac14+2^{k-1}$ $k = 1, 2, . . .$ Determine the distribution of $Y$.
As I can recall for probability generating function $g_y^{k}(1)=E(Y^k)$ and $g_y^{k}(0)/k!=P(Y=k)$.
So in this case $E(Y)=g_y'(1)=\frac 1 4$ and then what?