In the present article, we define the k-Narayana sequence of integer numbers. We study recurrence relations and some combinatorial properties of these numbers, and of the sum of their first n terms. These properties are derived from matrix methods. We also study some relations between the k-Narayana sequence and convolved k-Narayana sequence, and permanents and determinants of one type of Hessenberg matrix. Finally, we show how these sequences arise from a family of substitutions.
