a(n) = (n+2)*Catalan(n) - 1.

Terms

a(0) =1a(1) =2a(2) =7a(3) =24a(4) =83a(5) =293a(6) =1055a(7) =3860a(8) =14299a(9) =53481a(10) =201551a(11) =764217a(12) =2912167a(13) =11143499a(14) =42791039a(15) =164812364a(16) =636438059a(17) =2463251009a(18) =9552773999a(19) =37112526989a(20) =144410649239a(21) =562724141459