A047110 Array 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 no up-step crosses the line y=2x/3. (Thus a path crosses the line only at lattice points and on right-steps.).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 5, 4, 1, 1, 5, 9, 10, 9, 5, 1, 1, 6, 14, 19, 10, 14, 6, 1, 1, 7, 20, 33, 29, 24, 20, 7, 1, 1, 8, 27, 53, 62, 29, 44, 27, 8, 1, 1, 9, 35, 80, 115, 91, 73, 71, 35, 9, 1, 1, 10, 44, 115, 195, 206, 164, 144, 106, 44, 10, 1

Offset: 0

Author

Clark Kimberling. Definition revised Dec 08 2006

Keywords

• nonn
• tabl

Comments

T is the transpose of the array in A125778.

Examples

			Diagonals (beginning on row 0): {1}; {1,1}; {1,1,1}; {1,2,2,1};...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Cf. A125778.

Programs

Maple

T:= proc(h, k) option remember;
      `if`([h, k]=[0, 0], 1, `if`(h<0 or k<0, 0, T(h-1, k)+
      `if`(3*k>2*h and 3*(k-1)<2*h, 0, T(h, k-1))))
    end:
seq(seq(T(h,d-h), h=0..d), d=0..20); # Alois P. Heinz, Apr 04 2012

Mathematica

T[h_, k_] :=  T[h, k] = If[{h, k} == {0, 0}, 1, If[h<0 || k<0, 0, T[h-1, k]+If[3*k > 2*h && 3*(k-1) < 2*h, 0, T[h, k-1]]]]; Table[Table[T[h, d-h], {h, 0, d}], {d, 0, 20}] // Flatten (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)