a(n) = Sum_{k|n} mu(k)*Catalan(n/k) (mu = Moebius function A008683).

Terms

a(0) =1a(1) =1a(2) =4a(3) =12a(4) =41a(5) =126a(6) =428a(7) =1416a(8) =4857a(9) =16753a(10) =58785a(11) =207868a(12) =742899a(13) =2674010a(14) =9694799a(15) =35356240a(16) =129644789a(17) =477633711a(18) =1767263189a(19) =6564103612a(20) =24466266587a(21) =91482504853a(22) =343059613649