Augmentation of the Catalan triangle, A009766. See Comments.

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =3a(5) =3a(6) =1a(7) =6a(8) =14a(9) =14a(10) =1a(11) =10a(12) =41a(13) =86a(14) =86a(15) =1a(16) =15a(17) =95a(18) =327a(19) =645a(20) =645a(21) =1a(22) =21a(23) =190a(24) =965a(25) =2991a(26) =5662a(27) =5662a(28) =1a(29) =28