Array T(i,j) = binomial(-1/2-i,j)*(-4)^j, i,j >= 0 read by antidiagonals going down.

Terms

a(0) =1a(1) =2a(2) =1a(3) =6a(4) =6a(5) =1a(6) =20a(7) =30a(8) =10a(9) =1a(10) =70a(11) =140a(12) =70a(13) =14a(14) =1a(15) =252a(16) =630a(17) =420a(18) =126a(19) =18a(20) =1a(21) =924a(22) =2772a(23) =2310a(24) =924a(25) =198a(26) =22a(27) =1a(28) =3432a(29) =12012