On generating functions involving the square root of a quadratic polynomial
J. Integer Seq. 10(5), Article 07.5.2, 7 p., electronic only (2007)
Summary
Summary: 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 $\frac{1+Ax-\sqrt{1+2Ax+Bx^{2}}}{2Cx^{2}}$.
