a(n) = Catalan(n) + 2^n - 0^n.

Terms

a(0) =1a(1) =3a(2) =6a(3) =13a(4) =30a(5) =74a(6) =196a(7) =557a(8) =1686a(9) =5374a(10) =17820a(11) =60834a(12) =212108a(13) =751092a(14) =2690824a(15) =9727613a(16) =35423206a(17) =129775862a(18) =477900844a(19) =1767787478a(20) =6565168996a(21) =24468364172a(22) =91486757944a(23) =343068002258