Generalizing Bailey's generalization of the Catalan numbers (Q2719875)

The authors consider sequences \(a_1,a_2,\ldots ,a_{n+r}\) containing \(n\) elements equal to \(m-1\) (\(m>1\) is a given natural number) and \(r\) elements equal to \(-1\). Let \(\thickfracwithdelims\{\}\thickness 0nr_{m-1}\) be the number of the above sequences satisfying, in addition, the inequality \(a_1+a_2+\cdots +a_j\geq 0\) for each \(j=1,2,\ldots ,n+r\). An expression for \(\thickfracwithdelims\{\}\thickness 0nr_{m-1}\) is found. The case \(m=2\) was studied by \textit{D. F. Bailey} [Math. Mag. 69, 128-131 (1996; Zbl 0859.05007)]. The numbers \(\thickfracwithdelims\{\}\thickness 0nn_1\) coincide with the Catalan numbers. As a corollary, a new identity involving binomial coefficients is found.