In this article, we study the global stability and the asymptotic properties of the positive solutions of the nonlinear difference equation $$x_{n+1}=Ax_n+Bx_{n-k}+\dfrac{1+x_nx_{n-k}}{px_n+x_{n-k}}, \quad n=0,1,2,\dots $$ where the parameters $A, B, p$ and the initial conditions $x_{−k}, \dots, x_{−1}, x_0$ are arbitrary positive real numbers, while $k$ is a positive integer number. Some numerical examples will be given to illustrate our results.