Geodesic
Explicit Formula for the Generating Series of Diagonal 3D Rook Paths
Séminaire lotharingien de combinatoire, Tome 66 (2011-2012)

Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

Let an denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an n x n x n three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven 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:

$\displaystyle 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. $ 


@article{SLC_2011-2012_66_a0,
     author = {Alin Bostan and Fr\'ed\'eric Chyzak and Mark van Hoeij and Lucien Pech},
     title = {Explicit {Formula} for the {Generating} {Series} of {Diagonal} {3D} {Rook} {Paths}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {66},
     year = {2011-2012},
     url = {http://geodesic.mathdoc.fr/item/SLC_2011-2012_66_a0/}
}
TY  - JOUR
AU  - Alin Bostan
AU  - Frédéric Chyzak
AU  - Mark van Hoeij
AU  - Lucien Pech
TI  - Explicit Formula for the Generating Series of Diagonal 3D Rook Paths
JO  - Séminaire lotharingien de combinatoire
PY  - 2011-2012
VL  - 66
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2011-2012_66_a0/
ID  - SLC_2011-2012_66_a0
ER  - 
%0 Journal Article
%A Alin Bostan
%A Frédéric Chyzak
%A Mark van Hoeij
%A Lucien Pech
%T Explicit Formula for the Generating Series of Diagonal 3D Rook Paths
%J Séminaire lotharingien de combinatoire
%D 2011-2012
%V 66
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2011-2012_66_a0/
%F SLC_2011-2012_66_a0
Alin Bostan; Frédéric Chyzak; Mark van Hoeij; Lucien Pech. Explicit Formula for the Generating Series of Diagonal 3D Rook Paths. Séminaire lotharingien de combinatoire, Tome 66 (2011-2012). http://geodesic.mathdoc.fr/item/SLC_2011-2012_66_a0/