Example 2

The following sequence arose in the analysis of the long-term dynamics of numerical methods. For details on the mathematics of this sequence, see [2], but for now note that this could (broadmindedly) be considered as `applied mathematics' because RMC was investigating the reliability of numerical methods for solving nonlinear differential equations over long time intervals (the classical theory gives results useful only on compact time intervals, and the presence of exponentially growing terms in the classical error bounds raises questions about the validity of numerical solutions over long time intervals).

Define the function . Then multiplying each coefficient by k! we get the following sequence:

This (modulo the obviously trivial minus signs) is sequence M3024 in the book, which gives the reference to [3].

The history of the example is perhaps more interesting than the mathematics. The first few terms of a series representing the `modified equation' solved by , which arises from forward Euler applied to , were laboriously computed using Maple. Bruno Salvy's gfun package was then used to identify the sequence; it succeeded, but on checking it was found that the wrong sequence had been generated in the first place (i.e. there was a bug in my Maple program--RMC). Once the bug was fixed, gfun could no longer identify the sequence; Bruno Salvy (who is at INRIA in France) was asked for help, and he remarked (immediately) that he recognized the sequence. It turned out that he had a pre-publication version of the book under review here, and as stated previously the sequence is listed in the book! Coincidentally, Gilbert Labelle (from Montréal, the author of the reference [3]) was visiting INRIA at this time, as well, so it is conceivable that even without the book the sequence would have been recognized, but the book did play a rôle.

It is worth remarking that the paper by Labelle that was uncovered by this recognition was extremely apt, and would never have been discovered otherwise because it is unlikely in the extreme that RMC would have looked in a combinatorics journal for a result on reliability of numerical methods for dynamical systems.