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%I #20 Jul 10 2018 12:26:00
%S 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,
%T 99,1,8,33,95,194,293,293,1,9,42,137,331,624,624,1,10,52,189,520,1144,
%U 1768,1768,1,11,63,252,772,1916,3684,5452,5452
%N 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatticePath.html">Lattice Path</a>.
%F 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).
%e The table begins:
%e n=0: 1;
%e n=1: 1;
%e n=2: 1, 1;
%e n=3: 1, 2, 2;
%e n=4: 1, 3, 3;
%e n=5: 1, 4, 7, 7;
%e n=6: 1, 5, 12, 19, 19;
%e n=7: 1, 6, 18, 37, 37;
%e n=8: 1, 7, 25, 62, 99, 99;
%e n=9: 1, 8, 33, 95, 194, 293, 293.
%t 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]];
%t Table[T[n, k], {n, 0, 15}, {k, 0, Floor[2n/3]}] // Flatten (* _Jean-François Alcover_, Jul 10 2018 *)
%o (Sage)
%o T = [[1]]
%o for n in range(1,15):
%o T.append([T[-1][0]])
%o for k in range(1,floor(2*n/3) + 1):
%o T[-1].append(T[-1][k-1])
%o if 2*(n-1)>3*k:
%o T[-1][-1] += T[-2][k]
%Y Cf. A009766, A293946.
%K nonn,tabf
%O 0,6
%A _Danny Rorabaugh_, Oct 24 2017
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