a(n) = Sum_{k=0..floor(n/3)} binomial(n+2,3*k+2) * Catalan(k).

Terms

a(0) =1a(1) =3a(2) =6a(3) =11a(4) =21a(5) =42a(6) =86a(7) =180a(8) =387a(9) =852a(10) =1908a(11) =4329a(12) =9933a(13) =23013a(14) =53748a(15) =126386a(16) =298953a(17) =710847a(18) =1698086a(19) =4073181a(20) =9806565a(21) =23689555a(22) =57401322a(23) =139475190a(24) =339767545a(25) =829638396a(26) =2030206248a(27) =4978136133a(28) =12229451631a(29) =30095772966