Geodesic
On generating functions involving the square root of a quadratic polynomial
Journal of integer sequences, Tome 10 (2007) no. 5.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $\frac{1+Ax-\sqrt{1+2Ax+Bx^{2}}}{2Cx^{2}}$.
Classification : 05A15
Keywords: generating function, colored lattice path 
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     title = {On generating functions involving the square root of a quadratic polynomial},
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Callan, David. On generating functions involving the square root of a quadratic polynomial. Journal of integer sequences, Tome 10 (2007) no. 5. http://geodesic.mathdoc.fr/item/JIS_2007__10_5_a7/