Ascent sequences and the binomial convolution of Catalan numbers

In this paper, we consider two sets of pattern-avoiding ascent sequences: those avoiding both 201 and 210 and those avoiding 0021. In each case we show that the number of such ascent sequences is given by the binomial convolution of the Catalan numbers. The result for $\{201,210\}$-avoiders completes a family of results given by Baxter and the current author in a previous paper. The result for 0021-avoiders, together with previous work of Duncan, Steingr'{i}msson, Mansour, and Shattuck, completes the Wilf classification of single patterns of length 4 for ascent sequences.