A294207 Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,k), 0 <= 3k <= 2n, that are below the line 3y=2x, only touching at the end points.

1, 1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 4, 7, 7, 1, 5, 12, 19, 19, 1, 6, 18, 37, 37, 1, 7, 25, 62, 99, 99, 1, 8, 33, 95, 194, 293, 293, 1, 9, 42, 137, 331, 624, 624, 1, 10, 52, 189, 520, 1144, 1768, 1768, 1, 11, 63, 252, 772, 1916, 3684, 5452, 5452

Offset: 0

Author

Danny Rorabaugh, Oct 24 2017

Keywords

• nonn
• tabf

Examples

			The table begins:
n=0: 1;
n=1: 1;
n=2: 1, 1;
n=3: 1, 2,  2;
n=4: 1, 3,  3;
n=5: 1, 4,  7,  7;
n=6: 1, 5, 12, 19,  19;
n=7: 1, 6, 18, 37,  37;
n=8: 1, 7, 25, 62,  99,  99;
n=9: 1, 8, 33, 95, 194, 293, 293.
		

Links

• Eric Weisstein's World of Mathematics, Lattice Path.

Crossrefs

Cf. A009766, A293946.

Programs

Mathematica

T[, 0] = 1; T[n, k_] := T[n, k] = Which[0 < k < 2(n-1)/3, T[n-1, k] + T[n, k-1], 2(n-1) <= 3k <= 2n, T[n, k-1]];
Table[T[n, k], {n, 0, 15}, {k, 0, Floor[2n/3]}] // Flatten (* Jean-François Alcover, Jul 10 2018 *)

Sage

T = [[1]]
for n in range(1,15):
    T.append([T[-1][0]])
    for k in range(1,floor(2*n/3) + 1):
        T[-1].append(T[-1][k-1])
        if 2*(n-1)>3*k:
            T[-1][-1] += T[-2][k]

Formula

T(n,0) = 1; for 0 < k < 2(n-1)/3, T(n,k) = T(n-1,k) + T(n,k-1); for 2(n-1) <= 3k <= 2n, T(n,k) = T(n,k-1).