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.