a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(n-3*k).

Terms

a(0) =1a(1) =1a(2) =2a(3) =5a(4) =13a(5) =38a(6) =117a(7) =373a(8) =1222a(9) =4085a(10) =13877a(11) =47766a(12) =166229a(13) =583893a(14) =2067414a(15) =7371093a(16) =26440789a(17) =95355990a(18) =345538389a(19) =1257486165a(20) =4593933398a(21) =16841578325a(22) =61938532181a(23) =228454719830a(24) =844882459989