a(n) is the number of Dyck paths of semilength n+6 having its first peak at height n+1.

Terms

a(0) =42a(1) =132a(2) =297a(3) =572a(4) =1001a(5) =1638a(6) =2548a(7) =3808a(8) =5508a(9) =7752a(10) =10659a(11) =14364a(12) =19019a(13) =24794a(14) =31878a(15) =40480a(16) =50830a(17) =63180a(18) =77805a(19) =95004a(20) =115101a(21) =138446a(22) =165416a(23) =196416a(24) =231880a(25) =272272a(26) =318087a(27) =369852a(28) =428127a(29) =493506