a(n) = T(2n,n), where T(n,k) is in A037027.

Terms

a(0) =1a(1) =2a(2) =9a(3) =40a(4) =190a(5) =924a(6) =4578a(7) =22968a(8) =116325a(9) =593450a(10) =3045185a(11) =15699840a(12) =81260816a(13) =421993040a(14) =2197653240a(15) =11472991008a(16) =60023749566a(17) =314621200260