On the zeros of some polynomials with combinatorial coefficients

We consider two general classes of second-order linear recurrent sequences and the polynomials whose coefficients belong to a sequence in either of these classes. We show for each such sequence {a(i)}(i >= 0) that the polynomial f(x) = Sigma(n)(i=0) a(i)x(i) always has the smallest possible number of real zeros, that is, none when the degree is even and one when the degree is odd. Among the sequences then for which this is true are the Motzkin, Riordan, Schroder, and Delannoy numbers.