Abstract

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 \frac{1}{\sqrt{1−6x+x^{2}}} . Here we determine all generating functions of the form \frac{1}{\sqrt{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−\sqrt{1 + Ax + Bx^{2}} and \frac{1+Ax−\sqrt{1+2Ax+Bx^{2}}}{2Cx^{2}} .