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

Terms

a(0) =1a(1) =2a(2) =6a(3) =18a(4) =55a(5) =173a(6) =560a(7) =1858a(8) =6291a(9) =21657a(10) =75581a(11) =266797a(12) =950911a(13) =3417339a(14) =12369284a(15) =45052514a(16) =165002459a(17) =607283489a(18) =2244901889a(19) =8331383609a(20) =31030387439a(21) =115948830659a(22) =434542177289