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

Abstract

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: \[ G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.\]

To appear in "Séminaire Lotharingien de Combinatoire"

References in corpus (2)

• A Fast Approach to Creative Telescoping
• Differential Equations for Algebraic Functions

Cited by in corpus (7)

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• Telescopers for Rational and Algebraic Functions via Residues
• Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
• Some Open Problems related to Creative Telescoping