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Explicit formula for the generating series of diagonal 3D rook paths (1105.4456v2)
Published 23 May 2011 in cs.SC, cs.DM, and math.CO
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 $G(x)= \sum_{n \geq 0} a_n xn$ admits the following explicit expression in terms of a Gaussian hypergeometric function: [ G(x) = 1 + 6 \cdot \int_0x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)3}}}{(1-4w)(1-64w)} \, dw.]