Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)*C(2k,k)*(2k+1)/(n+k+1).

Terms

a(0) =1a(1) =3a(2) =1a(3) =10a(4) =4a(5) =2a(6) =35a(7) =15a(8) =9a(9) =5a(10) =126a(11) =56a(12) =36a(13) =24a(14) =14a(15) =462a(16) =210a(17) =140a(18) =100a(19) =70a(20) =42a(21) =1716a(22) =792a(23) =540a(24) =400a(25) =300a(26) =216a(27) =132a(28) =6435a(29) =3003