a(n) = T(n, floor(n/2)), where T = Catalan triangle (A008315).

Terms

a(0) =1a(1) =1a(2) =2a(3) =3a(4) =5a(5) =9a(6) =14a(7) =28a(8) =42a(9) =90a(10) =132a(11) =297a(12) =429a(13) =1001a(14) =1430a(15) =3432a(16) =4862a(17) =11934a(18) =16796a(19) =41990a(20) =58786a(21) =149226a(22) =208012a(23) =534888a(24) =742900a(25) =1931540a(26) =2674440a(27) =7020405a(28) =9694845a(29) =25662825