  Recently an expansion as a power series in 1/d has been presented for the
specific entropy of a complete dimer covering of a d-dimensional hypercubic
lattice. This paper extends from 3 to 10 the number of terms known in the
series. Likewise an expansion for the entropy, dependent on the dimer-density
p, of a monomer-dimer system, involving a sum sum_k a_k(d) p^k, has been
recently offered. We herein extend the number of the known expansion
coefficients from 6 to 20 for the hyper-cubic lattices of general dimension d
and from 6 to 24 for the hyper-cubic lattices of dimensions d < 5 . We show
that this extension can lead to accurate numerical estimates of the p-dependent
entropy for lattices with dimension d > 2. The computations of this paper have
led us to make the following marvelous conjecture: "In the case of the
hyper-cubic lattices, all the expansion coefficients, a_k(d), are positive"!
This paper results from a simple melding of two disparate research programs:
one computing to high orders the Mayer series coefficients of a dimer gas, the
other studying the development of entropy from these coefficients. An effort is
made to make this paper self-contained by including a review of the earlier
works.
