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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 $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: \[ G(x) = 1 + 6 \cdot \int_0^x \frac{ \,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw. \]
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Submitted on : Wednesday, January 23, 2013 - 10:31:26 PM
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Alin Bostan, Frédéric Chyzak, Mark Van Hoeij, Lucien Pech. Explicit formula for the generating series of diagonal 3D rook paths. Seminaire Lotharingien de Combinatoire, Université Louis Pasteur, 2011, 66, pp.1-27. ⟨hal-00780432⟩

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