  In this case study in ``fully automated enumeration'', we illustrate how to
take full advantage of symbolic computation by developing (what we call)
`symbolic-dynamical-programming' algorithms for computing many terms of `hard
to compute sequences', namely the number of Latin trapezoids, generalized
derangements, and generalized three-rowed Latin rectangles. At the end we also
sketch the proof of a generalization of Ira Gessel's 1987 theorem that says
that for any number of rows, k, the number of Latin rectangles with k rows and
n columns is P-recursive in n. Our algorithms are fully implemented in Maple,
and generated quite a few terms of such sequences.
