How to find/guess a polynomial sequence? My question is motivated by the recent question and more recent
appearance of its author Bruce Westbury.
Most of you know that the best way to find a sequence of integers is looking for it on
The On-Line Encyclopedia of Integer Sequences.
(My personal success of using this powerful database is however less than 10%.)
Another strategy (especially, when you suspect that your sequence is
holonomic) would be trying
a guessing package in a computer algebra system, like $\operatorname{gfun}$ in Maple.
What can we do if we need to identify a sequence of polynomials?
(for simplicity with integer coefficients)
One recipe (which was used by my colleague for solving the problem in
this question) is again
to use the OEIS, since the latter contains many 2D examples as well
(like the whole Pascal triangle of binomial coefficients and several
subcollections from it). The chances are miserable (as Bruce's sequence shows).
Even having some additional information (like knowing that the polynomials
are $q$-analogues of a known integer sequence), there seems to be no general
machinery or database to assist in identification.
Are there algorithms (better implemented) for polynomials analogous to $\operatorname{gfun}$?
Thanks!
P.S. The tag "soft-question" here means that the question is indeed soft
but also on software.
 A: You do realize that gfun works for polynomials as well as for integer sequences?  The version that ships with Maple is not always the most up-to-date however, for that you should get it from the source.  I know the authors very well, and the implementation of the guessing algorithms in gfun are state-of-the-art.  [That doesn't mean that they can't be improved!]  I know that they like to receive feedback and bug reports.
For those who are curious: the core algorithm behind gfun's guessing capabilities is simultaneous Hermite-Pade approximation, using a variant of the Beckermann-Labahn algorithm.
A: An obvious way to convert a sequence of polynomials to a sequence of numbers is to plug in a number.  Small integers, ..., -1, 0, 1, ... often give illuminating sequences, and I have many times guessed a sequence of polynomials by this method, either by recognizing that the resulting sequence was holonomic or by finding it in Sloane.  For sequences of $q$-series, amazing things sometimes happen when you plug in a root of unity.  For example, in Bruce Westbury's sequence that inspired the question, plugging in $e^{2\pi i/6}$ produces only the integers -1, 0, 1 for the terms listed in his question.  Plugging in $e^{2\pi i/5}$ doesn't always give integers, but does give 0 for the last three terms.  I don't know how to use this to guess the sequence, but it is curious.  I wonder if it continues...
A: Sounds like a problem you'd find in the book A=B.
