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Mean Coulomb energy of 2d Dyson gas in quadratic potential is examined fromcombinatorial viewpoint. For beta = 1, we find a recursive relation on meanenergy and obtain its exact (finite N) solution in closed form in terms of thehypergeometric function 3F2. Using this exact solution, we derive the large-Nasymptotic expansion of mean energy and show, that this expansion containshalf-integer powers of N.

Exact solution for mean energy of 2d Dyson gas at beta = 1