New approximations for the higher order coefficients in an asymptotic expansion for the Barnes $G$-function

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".