a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(n-3*k).

Terms

a(0) =1a(1) =1a(2) =2a(3) =5a(4) =15a(5) =46a(6) =147a(7) =485a(8) =1642a(9) =5669a(10) =19883a(11) =70646a(12) =253755a(13) =919925a(14) =3361546a(15) =12368661a(16) =45786219a(17) =170400470a(18) =637200555a(19) =2392962645a(20) =9021255722a(21) =34128098389a(22) =129519490219a(23) =492966689110