a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).

Terms

a(0) =1a(1) =0a(2) =-1a(3) =3a(4) =0a(5) =-5a(6) =15a(7) =0a(8) =-28a(9) =84a(10) =0a(11) =-165a(12) =495a(13) =0a(14) =-1001a(15) =3003a(16) =0a(17) =-6188a(18) =18564a(19) =0a(20) =-38760a(21) =116280a(22) =0a(23) =-245157a(24) =735471a(25) =0a(26) =-1562275a(27) =4686825a(28) =0a(29) =-10015005