  The virial expansion characterizes the high-temperature approach to the
quantum-classical crossover in any quantum many-body system. Here, we calculate
the virial coefficients up to the fifth-order of Fermi gases in 1D, 2D, and 3D,
with attractive contact interactions, as relevant for a variety of applications
in atomic and nuclear physics. To that end, we discretize the imaginary-time
direction and calculate the relevant canonical partition functions. In coarse
discretizations, we obtain analytic results featuring relationships between the
interaction-induced changes $\Delta b_3$, $\Delta b_4$, and $\Delta b_5$ as
functions of $\Delta b_2$, the latter being exactly known in many cases by
virtue of the Beth-Uhlenbeck formula. Using automated-algebra methods, we push
our calculations to progressively finer discretizations and extrapolate to the
continuous-time limit. We find excellent agreement for $\Delta b_3$ with
previous calculations in all dimensions and we formulate predictions for
$\Delta b_4$ and $\Delta b_5$ in 1D and 2D. We also provide, for a range of
couplings,the subspace contributions $\Delta b_{31}$, $\Delta b_{22}$, $\Delta
b_{41}$, and $\Delta b_{32}$, which determine the equation of state and static
response of polarized systems at high temperature. As a performance check, we
compare the density equation of state and Tan contact with quantum Monte Carlo
calculations, diagrammatic approaches, and experimental data where available.
Finally, we apply Pad\'e and Pad\'e-Borel resummation methods to extend the
usefulness of the virial coefficients to approach and in some cases go beyond
the unit-fugacity point.
