<p>Sylvester–Kac type matrices are tridiagonal integral matrices with integral spectra or with eigenvalues presenting some kind of closed form. In general, the entries of the two non-zero subdiagonals are assumed to be strictly positive. There has been a particular interest on the quadratic spectra or some kind of products involving the entries of the subdiagonals. In this article we develop a new combinatorial approach based on a lower Pascal’s triangle matrix, which turns out to be a much more general approach to investigate the Sylvester–Kac matrices. As a result, infinitely many Sylvester–Kac type matrices can be produced. Several new examples with integral spectra are provided as well.</p>