a(n) = binomial(n, floor((n-1)/2))^2.

Terms

a(0) =0a(1) =1a(2) =1a(3) =9a(4) =16a(5) =100a(6) =225a(7) =1225a(8) =3136a(9) =15876a(10) =44100a(11) =213444a(12) =627264a(13) =2944656a(14) =9018009a(15) =41409225a(16) =130873600a(17) =590976100a(18) =1914762564a(19) =8533694884a(20) =28210561600a(21) =124408576656a(22) =418151049316