a(n) = (1/2)*(1 + Sum_{k=0..n} binomial(2*k, k)).

Terms

a(0) =1a(1) =2a(2) =5a(3) =15a(4) =50a(5) =176a(6) =638a(7) =2354a(8) =8789a(9) =33099a(10) =125477a(11) =478193a(12) =1830271a(13) =7030571a(14) =27088871a(15) =104647631a(16) =405187826a(17) =1571990936a(18) =6109558586a(19) =23782190486a(20) =92705454896a(21) =361834392116