@article{BoChHoPe11,
    author      = {Bostan, Alin and Chyzak, Fr\'ed\'eric and van Hoeij, Mark and Pech, Lucien},
    title       = {Explicit formula for the generating series of diagonal {3D} rook paths},
    journal     = {S\'eminaire Lotharingien de Combinatoire},
    year        = {2011},
    volume      = {B66a},
    page        = {27 pages},
    url         = {http://www.emis.de/journals/SLC/wpapers/s66bochhope.html},
    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: \[ G(x) =  1 + 6 \cdot \int_0^x \frac{ \,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw. \]},
}