Explicit Formula for the Generating Series of Diagonal 3D Rook Paths

Abstract. Let an denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an n x n x n three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven discovery and proof of the fact that the generating series admits the following explicit expression in terms of a Gaussian hypergeometric function:

Received: May 18, 2011. Accepted: July 7, 2011. Final Version: October 4, 2011.

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