Explicit formula for the generating series of diagonal 3D rook paths

Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n\times n\times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \emph{discovery and proof} of the fact that the generating series $G(x)={\sum }_{n\ge 0}{a}_n{x}^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: $$G(x)=1+6\cdot {\int }_0^x\frac{\text{pFq}211\mathrm{/}32\mathrm{/}32\frac{27w(2-3w)}{(1-4w{)}^3}}{(1-4w)(1-64w)}dw\mathrm{.}$$