a(n) = floor(C(n/2)*C(n/2+1)), where C = Catalan numbers (A000108).

Terms

a(0) =1a(1) =1a(2) =2a(3) =4a(4) =10a(5) =25a(6) =70a(7) =199a(8) =588a(9) =1784a(10) =5544a(11) =17569a(12) =56628a(13) =185202a(14) =613470a(15) =2054998a(16) =6952660a(17) =23732911a(18) =81662152a(19) =283026021a(20) =987369656a(21) =3465222945a(22) =12228193432a(23) =43369190282a(24) =154532114800a(25) =552998717472