a(n) = Catalan(n) - Motzkin(n-1).

Terms

a(0) =0a(1) =1a(2) =3a(3) =10a(4) =33a(5) =111a(6) =378a(7) =1303a(8) =4539a(9) =15961a(10) =56598a(11) =202214a(12) =727389a(13) =2632605a(14) =9581211a(15) =35047098a(16) =128791323a(17) =475281921a(18) =1760726808a(19) =6545921136a(20) =24415415001a(21) =91340016081a(22) =342658850427