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

by Alin Bostan, Frédéric Chyzak, Mark van Hoeij, and Lucien Pech

(September 2011)

*All Maple code intended for Maple13.*

- 3D diagonal rook sequence in Sloane's OEIS.
- 3D diagonal queen sequence in Sloane's OEIS.
- Maple code of an optimised implementation of Lipshiz's approach (Section 2.3).
- Maple code to produce the
operator
*P*and rational-function certificates*S*and*T*for the 3D rooks (Section 2.4). - Corresponding rational-function
certificates
*S*and*T*. (For comparison sake, here is another set of certificates, which was obtained with an earlier piece of code: observe the different denominator structure.) - The version of Mgfun used for the calculations above (version 4.0) is part of Algolib 13.0.
- Maple code to get the minimal-order recurrence for the 3D rooks (Section 2.5).
- Maple code to get the explicit form for the generating series (Section 2.6).
- Program equiv used in the calculation above.
- Comparison between the explicit formula obtained by creative telescoping and that obtained by the approach of Frits Beukers (Section 3.3).
- Guessed linear differential equation for the 3D queens, together with the corresponding linear recurrence equation (Section 3.4).
- Maple code to get the rectangular system for the 3D rooks (Section A.1).
- Maple code to perform the second iteration of Chyzak's algorithm for the 3D rooks (Section A.2).
- Maple code to reconstruct the final certificates from Stages A and B in the case of the 3D rooks (Section A.3).