A rectangular Vandermonde matrix \(V=\{V_{ij}\}= \{x_i^{j-1}\}\) \((i=1,\dots, n;\;j=1,\dots, m;\;n\leq m)\) defined on the so-called Chebyshev nodes (the roots of Chebyshev polynomials of the first order) is studied, by making use of combinatorial identities from number theory [cf. \textit{A. Eisinberg}, \textit{P. Pugliese}, and \textit{N. Salerno}, Numer. Math. 87, No. 4, 663--674 (2001; Zbl 0974.65029); \textit{A. Eisinberg}, \textit{G. Franzé}, and \textit{P. Puyliese}, Linear Algebra Appl. 283, No. 1-3, 205--219 (1998; Zbl 0935.65016) and Numer. Math. 80, No. 1, 75--85 (1998; Zbl 0913.65022)]. The explicit factorizations \(V=HUD\) as well as \(V^+= (1/n)D^{-1}QBH^T\) are presented, where \(H\) is rectangular, \(U\) and \(Q=U^{-1}\) are triangular, \(D\) and \(B\) \((B_{1,1}= 1;\;B_{i,i}=2;\;i=2,\dots,n)\) are diagonal matrices. It is proved that the so-called condition number [cf. \textit{G. H. Golub} and \textit{C. F. Van Loan}, Matrix computations, 2nd ed. (1989; Zbl 0733.65016), 1. Aufl. (Zbl 0559.65011)] of the matrix \(V\) does not depend on the dimension of the nodes sets. Numerical experiments are also performed and, as a result, some numerical properties of the proposed formulae are established.