Many familiar counting sequences, such as the Catalan, Motzkin,
Schröder and Delannoy numbers, have a generating function
that is algebraic of degree 2. For example, the GF for the
central Delannoy numbers is
. Here
we determine all generating functions
of the form
that yield counting sequences and
point out that they have a unified combinatorial
interpretation in terms of colored lattice paths. We do likewise
for the related forms
and
.
. Here
we determine all generating functions
of the form
that yield counting sequences and
point out that they have a unified combinatorial
interpretation in terms of colored lattice paths. We do likewise
for the related forms
and
.