We define a certain class of polynomials denoted by P_n,m,p(x), and give the combinatorial meaning of the coefficients. Chebyshev polynomials are special cases of P_n,m,p(x). We first show that P_n,m,p(x) can be expressed in terms of P_n,0,p(x). From this we derive that P_n,2,2(x) can be obtained in terms of trigonometric functions, from which we obtain some of its important properties. Some questions about orthogonality are also addressed. Furthermore, it is shown that P_n,2,2(x) fulfills the same three-term recurrence as the Chebyshev polynomials. We also obtain some other recurrences for P_n,m,p(x) and its coefficients. Finally, we derive a formula for the coefficients of Chebyshev polynomials of the second kind.