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We compute the free energy of the planar monomer-dimer model. Unlike theclassical planar dimer model, an exact solution is not known in this case. Eventhe computation of the low-density power series expansion requires heavy andnontrivial computations. Despite of the exponential computational complexity,we compute almost three times more terms than were previously known. Such anexpansion provides both lower and upper bound for the free energy, and allowsto obtain more accurate numerical values than previously possible. We expectthat our methods can be applied to other similar problems.

Power series expansions for the planar monomer-dimer problem