  In this paper, firstly, by a determinant of deformed Pascal's triangle,
namely the normalized Hessenberg matrix determinant, to count Dyck paths, we
give another combinatorial proof of the theorems which are of Catalan numbers
determinant representations and the recurrence formula. Secondly, a determinant
of normalized Toeplitz-Hessenberg matrix, whose entries are binomials, arising
in power series, we derive new four properties of Pascal's triangle.
