  In this paper, we provide new formulas for determining the coefficients
appearing in the asymptotic expansion for the Barnes $G$-function as $n$ tends
to infinity for certain classes of asymptotic expansion for the Barnes
$G$-function. We remark that our formulas can be used to approximate the
coefficients appearing in an asymptotic expansion of the ``random matrix
factor" from the Keathing-Snaith conjecture and the coefficients appearing in
an asymptotic expansion of the ``L\'evy-Khintchine type representation of the
reciprocal of the Barnes $G$-function".
