What does term per term multiplication of power series look like?

Question

Suppose we have two rational functions $f$ and $g$ on the complex plane, such that $f(0)=g(0)=1$. In a neighbourhood of $0$, we can write the following. \begin{align} f(t) = \sum_{n=0}^{\infty} N_n^f t^n \\ g(t) = \sum_{n=0}^{\infty} N_n^g t^n \end{align} Now my question is, what does the following function look like? \begin{align} h(t) := \sum_{n=0}^{\infty} N_n^f N_n^g t^n \end{align} Evidently, we have $h(0)=1$ and \begin{align} m! \left. \frac{d^m h}{dt^m} \right|_{t=0} = \left. \frac{d^m f}{dt^m} \right|_{t=0} \cdot \left. \frac{d^m g}{dt^m} \right|_{t=0} \end{align} However, I would like to have a formula that gives $h$ in terms of $f$ and $g$ explicitly, without having to refer to either the derivatives of $h$, or a series expansion.