A047100 Array T read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and touches the line y=x/3 only at lattice points.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 4, 6, 5, 3, 1, 1, 5, 10, 11, 5, 4, 1, 1, 6, 15, 21, 16, 5, 5, 1, 1, 7, 21, 36, 37, 21, 10, 6, 1, 1, 8, 28, 57, 73, 58, 31, 16, 7, 1, 1, 9, 36, 85, 130, 131, 89, 31, 23, 8, 1, 1, 10, 45, 121, 215, 261, 220