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

In this article, we solve a problem of enumerative combinatorics addressed and left open in [16]. The initial question is formulated in terms of paths on an infinite three-dimensional chessboard. The 3D chessboard being identified with N, a 3D rook is a piece which is allowed to move parallelly to one of the three axes. The general objective is to count paths (i.e., finite sequences of moves) of a 3D rook on the 3D chessboard. Following [16], we further restrict to 3D rook paths starting from the cell (0, 0, 0), whose steps are positive integer multiples of (1, 0, 0), (0, 1, 0), or (0, 0, 1). In other words, we assume that the piece moves closer to the goal cell at each step. For example, one such rook path is