On Generating Functions Involving the Square Root of a Polynomial
Many familiar counting sequences, such as the Catalan, Motzkin,
Schröder and Delannoy numbers, have a generating function (GF)
that is algebraic of degree 2. For example, the GF for the
central Delannoy numbers is 1/Ö(1-6x+x^2})). Here
we characterize GFs of the form 1/Ö(1+Ax+Bx^2))
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 1-Ö(1+Ax+Bx^2) and
(1+Ax-Ö(1+2Ax+Bx^2))/(2Cx^2).
