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

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

2011-05-23

arxiv.org/abs/1105.4456v2

Computer Science

Symbolic Computation

To appear in "S\'eminaire Lotharingien de Combinatoire"

Scientific paper

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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