D_{2,k}(n) = Sum_{i=0..n} binomial(n,i) * T_{2,k} (n-i), where T_{2,k}(n) = Sum_{L=0..n} binomial(n,L) * f_k(2n-L), and f_k(2n-L) is the number of k-noncrossing matchings over 2n-L vertices. [edited, using formulas from p. 12 of the paper by Chen et al., by Jon E. Schoenfield, Jul 21 2019]