a(n) = C(n)*(7*n + 1) where C(n) = Catalan numbers (A000108).

Terms

a(0) =1a(1) =8a(2) =30a(3) =110a(4) =406a(5) =1512a(6) =5676a(7) =21450a(8) =81510a(9) =311168a(10) =1192516a(11) =4585308a(12) =17681020a(13) =68346800a(14) =264769560a(15) =1027653570a(16) =3995416710a(17) =15557374800a(18) =60660114900a(19) =236813267460a(20) =925540979220