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 $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: [ G(x) = 1 + 6 \cdot \int_0^x \frac{,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} , dw.]

💡 Research Summary

The paper addresses the enumeration of diagonal rook paths on an n × n × n three‑dimensional chessboard. A rook may move any positive integer number of squares along one of the three coordinate axes, but each step must bring it strictly closer to the opposite corner (n,n,n). Let aₙ denote the number of such paths from (0,0,0) to (n,n,n). The authors aim to determine the ordinary generating function G(x)=∑_{n≥0}aₙxⁿ in a closed form.

Combinatorial formulation.
The set of admissible steps is S={ (k,0,0), (0,k,0), (0,0,k) | k≥1 }. The multivariate generating function for unrestricted walks with steps from S is

F(s,t,u)=1/(1−∑_{(i,j,k)∈S}s^i t^j u^k)=1/