The authors obtain sufficient conditions to assure that all the solutions to the equation \(x_{n+1}-x_n+p_n x_{n-k}=0\), \(n=0,1, 2, \ldots\) are oscillatory. They allow functions \(p_n\) to be nonoscillatory. The obtained results improve some previous results given for the case \(p_n \geq 0\) for \(n\) large enough. They present an example to demonstrate the advantage of the given results.