a(n) = 3*Catalan(n) - Catalan(n-1) - 1.

Terms

a(0) =1a(1) =4a(2) =12a(3) =36a(4) =111a(5) =353a(6) =1154a(7) =3860a(8) =13155a(9) =45525a(10) =159561a(11) =565249a(12) =2020687a(13) =7280419a(14) =26410094a(15) =96378164a(16) =353576699a(17) =1303271309a(18) =4824150869a(19) =17925098069a(20) =66834680639a(21) =249981423899a(22) =937696277309