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Single-file diffusion is a paradigm for strongly correlated classical stochastic many-body dynamics and has widespread applications in soft condensed matter and biophysics. However, exact results for {single-file} systems are sparse and limited to the simplest scenarios. We present an algorithm for computing the non-Markovian time-dependent conditional probability density function of a {tagged-particle} in a {single-file} of $N$ particles diffusing in a confining external potential. The algorithm implements an eigenexpansion of the full interacting many-body problem obtained by means of the coordinate Bethe ansatz. While formally exact, the Bethe eigenspectrum involves the generation and evaluation of permutations, {which becomes unfeasible for single-files with an increasing number of particles $N$.} Here we exploit the underlying {exchange} symmetries between the particles to the left and to the right of the {tagged-particle} and show that it is possible to reduce the complexity of the algorithm from the worst case scenario $\mathcal{O}(N!)$ down to $\mathcal{O}(N)$. A C++ code to calculate the non-Markovian probability density function using this algorithm is provided. Solutions for simple model potentials are readily implemented incl. {single-file diffusion} in a flat and a 'tilted' box, as well as in a parabolic potential. Notably, the program allows for implementations of solutions in arbitrary external potentials under the condition that the user can supply solutions to the respective single-particle eigenspectra.