a(n) = binomial(n, floor(n/2))^2 = A001405(n)^2.

Terms

a(0) =1a(1) =1a(2) =4a(3) =9a(4) =36a(5) =100a(6) =400a(7) =1225a(8) =4900a(9) =15876a(10) =63504a(11) =213444a(12) =853776a(13) =2944656a(14) =11778624a(15) =41409225a(16) =165636900a(17) =590976100a(18) =2363904400a(19) =8533694884a(20) =34134779536a(21) =124408576656a(22) =497634306624