a(n) = Sum_{k=0..floor(n/2)} Catalan(k).

Terms

a(0) =1a(1) =1a(2) =2a(3) =2a(4) =4a(5) =4a(6) =9a(7) =9a(8) =23a(9) =23a(10) =65a(11) =65a(12) =197a(13) =197a(14) =626a(15) =626a(16) =2056a(17) =2056a(18) =6918a(19) =6918a(20) =23714a(21) =23714a(22) =82500a(23) =82500a(24) =290512a(25) =290512a(26) =1033412a(27) =1033412a(28) =3707852a(29) =3707852