Table of notation.

We develop an efficient algorithm to find optimal observation times by maximizing the Fisher information for the birth rate of a partially observable pure birth process involving n observations. Partially observable implies that at each of the observation time points for counting the number of individuals present in the pure birth process, each individual is observed independently with a fixed probability p, modeling detection difficulties or constraints on resources. We apply concepts and techniques from generating functions, using a combination of symbolic and numeric computation, to establish a recursion for evaluating and optimizing the Fisher information. The recursion, while still computationally intensive, greatly improves on previously known computational methods which quickly became intractable even in the n = 2 case. Our numerical results reveal the efficacy of this new method. An implementation of the algorithm is available publicly.