a(n) = [x^n] (1 + x)/(1 - x)^(2*n+1).

Terms

a(0) =1a(1) =4a(2) =20a(3) =112a(4) =660a(5) =4004a(6) =24752a(7) =155040a(8) =980628a(9) =6249100a(10) =40060020a(11) =258048960a(12) =1668903600a(13) =10829900592a(14) =70480305440a(15) =459823234112