Here is a passage from Drake's applied probability book:
Let $p_x$ be the Probability Mass Function of a random variable $x$, we define the z-transform of $p_x$ as $p_x^T(z)$: $ p_x^T(z) = p_x(0) + zp_x(1) + z^2p_x(2) + \cdots $ then is it possible to determine the individual terms of the PMF from $p_x^T(z)$ by the following formula? If yes, why? $ p_x(x_0) = \frac{1}{x_0 !} \left[ \frac{d^{x_0}}{dz^{x_0}} p_x^T(z) \right]_{z=0} $
For instance, $ p_x(3) = \frac{1}{3 !} \left[ \frac{d^3}{dz^3} p_x^T(z) \right]_{z=0} $