A method for obtaining the algebraic generating function from a series

TL;DR

This paper introduces an experimental method utilizing integer relation algorithms and symbolic computation tools to approximate the algebraic generating function from initial series terms, aiding in the discovery of closed-form solutions.

Contribution

It presents a novel approach combining integer relations and symbolic computation to find algebraic generating functions from limited series data.

Findings

Successfully tested on known integer sequences

Provides a practical tool for conjecturing generating functions

Applicable to sequences in combinatorics and algebra

Abstract

We describe here an experimental method that permits to compute a good candidate for the closed form of a generating function if we know the first few terms of a series. The method is based on integer relations algorithms and uses either two programs of symbolic computation: Maple or Pari-Gp. Some results are presented in the appendix. This method was tested on a set of sequences that were part of the incoming book on integer sequences (as of 1993). This method was presented at the FPSAC, Formal Power Series and Algebraic Combinatorics, Florence, June 1993.