Binomial(n-k,k)^2 where k = ceiling(n/4).

Terms

a(0) =1a(1) =0a(2) =1a(3) =4a(4) =9a(5) =9a(6) =36a(7) =100a(8) =225a(9) =400a(10) =1225a(11) =3136a(12) =7056a(13) =15876a(14) =44100a(15) =108900a(16) =245025a(17) =627264a(18) =1656369a(19) =4008004a(20) =9018009a(21) =25050025a(22) =64128064a(23) =153165376a(24) =344622096a(25) =1012766976a(26) =2538950544a(27) =6009350400a(28) =13521038400a(29) =41408180100