Geodesic
On generating functions involving the square root of a quadratic polynomial
Journal of integer sequences, Tome 10 (2007) no. 5
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 
@article{JIS_2007__10_5_a7,
     author = {Callan,  David},
     title = {On generating functions involving the square root of a quadratic polynomial},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {5},
     zbl = {1138.05300},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_5_a7/}
}
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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/