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

Terms

a(0) =1a(1) =3a(2) =9a(3) =29a(4) =98a(5) =342a(6) =1221a(7) =4433a(8) =16302a(9) =60554a(10) =226746a(11) =854658a(12) =3239044a(13) =12332140a(14) =47137005a(15) =180780345a(16) =695367510a(17) =2681600130a(18) =10364759790a(19) =40142121030a(20) =155748675420a(21) =605274171060