We present an explicit formula for the transition matrix $\mathcal{ C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{ (1^r)}(x\,|\,a,-a,c,-c\,|\,q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{ (1^{ r})}(x)$. The entries of the matrix $\mathcal{ C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P^{ (C_n,C_n)}_{ (1^r)}(x\,|\,b;q,t)= P_{ (1^r)}\big(x\,|\,b^{ 1/2},-b^{ 1/2},q^{ 1/2}b^{ 1/2},-q^{ 1/2}b^{ 1/2}\,|\,q,t\big)$. It is also shown that the $q$-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials $P^{ (C_n,C_n)}_{ (1^r)}(x\,|\,q;q,q)$ to the Hall-Littlewood polynomials $P^{ (C_n,C_n)}_{ (1^r)}(x\,|\,t;0,t)$.
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