Tribinomial coefficients and Catalan numbers

摘要

The concept of the ordinary binomial coefficientn! n(n — 1) ... (n — r + 1)r! (n — r)!r (r — 1) ... 2 x 1can be employed to construct an interesting family of positive integers. Sucha family was introduced around 1974 by W. Hansell [1] using the triangularnumbers t_m= 1/2m(m +1), wherem≥1; we call themtribinomial coefficientssince they are binomial coefficients for triangular numbers. To this end, firstwe define t_m~* =t_2t_1,corresponding tom! = m(m — 1)...2 x 1, andt_0~*=1. For example,t_3~* = t_3t_2t_1=6 x 3 x 1 = 18.n[Tribinomial coefficientsare defined as follows:r[1:7tntn — 1... to— r +1(1)nr, — tX-r —trtr_i... t2ti