From a differential equation.

Terms

a(0) =1a(1) =3a(2) =6a(3) =11a(4) =24a(5) =69a(6) =227a(7) =753a(8) =2451a(9) =8004a(10) =27138a(11) =97806a(12) =375313a(13) =1511868a(14) =6292884a(15) =26826701a(16) =116994453a(17) =523646202a(18) =2414394601a(19) =11487130362a(20) =56341183365a(21) =284110648983