Suppose thatf(z) =g(z)/znwheregis a real entire function of finite order withg(0)≠ 0 andnis a positive integer, and thatff′, andf″ have only red zeros. We prove thatthengis a polynomial of degree not exceedingn+1. This strengthens an earlier result of the author, when one had to consider derivatives up to an order depending onfand the order of the entire functiongConversely, iffis of this form wheregis a polynomial of degree at mostnwith only real zeros, thenf(k) has only real zeros for allk≥ 0. If the degree ofgisn+1 thenf(k) has only real zeros for allk≥ 0 if, and only iffandf′ have only real zeros.