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

Terms

a(0) =1a(1) =1a(2) =0a(3) =0a(4) =0a(5) =5a(6) =14a(7) =42a(8) =90a(9) =207a(10) =407a(11) =858a(12) =1638a(13) =3341a(14) =6330a(15) =12750a(16) =24174a(17) =48467a(18) =92207a(19) =184566a(20) =352506a(21) =705201a(22) =1351825a(23) =2703880a(24) =5200000a(25) =10400275a(26) =20057949a(27) =40116222a(28) =77558354a(29) =155117085