Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.2

On Generating Functions Involving the Square Root of a Quadratic Polynomial

David Callan
Department of Statistics
University of Wisconsin-Madison
1300 University Avenue
Madison, WI 53706-1532


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 $\frac{1+Ax-\sqrt{1+2Ax+Bx^{2}}}{2Cx^{2}}$.

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(Concerned with sequences A000108 A000984 A001003 A001006 A001700 A001850 A002212 A002426 A003645 A005572 A006139 A006318 A006442 A007564 A025235 A026375 A059231 A059304 A068764 A069835 A071356 A080609 A081671 A084601 A084603 A084609 A084768 A084771 A084773 A090442 A098430 A098443 A098658 A098659 A101601 A107264 and A107265 .)

Received September 24 2006; revised version received May 4 2007. Published in Journal of Integer Sequences, May 7 2007.

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