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

0,6

Links

Table of n, a(n) for n=0..60.

Eric Weisstein's World of Mathematics, Lattice Path.

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).

Example

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.

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 *)

Prog

(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]

Crossrefs

Cf. A009766, A293946.

Sequence in context: A118816 A283904 A097289 * A288267 A237840 A114115

Adjacent sequences: A294204 A294205 A294206 * A294208 A294209 A294210

Keyword

nonn,tabf

Author

Danny Rorabaugh, Oct 24 2017

Status

approved