Several identities in the Catalan triangle

In this paper, we first establish several identities for the alternating sums in the Catalan triangle whose (n, p) entry is defined by Bn, p = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tfrac{p} {n}\left( {_{n - p}^{2n} } \right) $$\end{document}. Second, we show that the Catalan triangle matrix C can be factorized by C = FY = ZF, where F is the Fibonacci matrix. From these formulas, some interesting identities involving Bn, p and the Fibonacci numbers Fn are given. As special cases, some new relationships between the well-known Catalan numbers Cn and the Fibonacci numbers are obtained, for example: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_n = F_{n + 1} + \sum\limits_{k = 3}^n {\left\{ {1 - \frac{{(k + 1)(k5 - 6)}} {{4(2k - 1)(2k - 3)}}} \right\}C_k F_{n - k + 1} } , $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{gathered} \frac{{n - 1}} {{n + 2}}C_n = \frac{1} {2}F_n + F_{n - 2} \hfill \\ + \sum\limits_{k = 4}^n {\left\{ {1 - \frac{{(k + 2)(5k^2 - 16k + 9)}} {{4(k - 1)(2k - 1)(2k - 3)}}} \right\}\frac{{k - 1}} {{k + 2}}C_k F_{n - k + 1} } . \hfill \\ \end{gathered} $$\end{document}