In this article we give a combinatorial formula for a certain class of Koornwinder polynomials, also known as Macdonald polynomials of type C. In particular, we give a combinatorial formula for the Koornwinder polynomials K λ = K λ (z1, . . . , zN ; a, b, c, d; q, t), where λ = (1, . . . , 1, 0, . . . , 0). We also give combinatorial formulas for all "open boundary ASEP polynomials" Fµ, where µ is a composition in {-1, 0, 1} N ; these polynomials are related to the nonsymmetric Koornwinder polynomials Eµ up to a triangular change of basis. Our formulas are in terms of rhombic staircase tableaux, certain tableaux that we introduced in previous work to give a formula for the stationary distribution of the two-species asymmetric simple exclusion process (ASEP) on a line with open boundaries. Contents 1. Introduction 1 2. The Hecke algebra and Koornwinder polynomials 10 3. The proof of Theorem 1.10 16 4. Conclusion 24