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 .
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
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 )
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.