a(n) = Catalan(n) + 2^n - n - 1.

Terms

a(0) =1a(1) =1a(2) =3a(3) =9a(4) =25a(5) =68a(6) =189a(7) =549a(8) =1677a(9) =5364a(10) =17809a(11) =60822a(12) =212095a(13) =751078a(14) =2690809a(15) =9727597a(16) =35423189a(17) =129775844a(18) =477900825a(19) =1767787458a(20) =6565168975a(21) =24468364150a(22) =91486757921a(23) =343068002234