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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.

%I A294207 #21 Feb 16 2025 08:33:51
%S A294207 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 A294207 99,1,8,33,95,194,293,293,1,9,42,137,331,624,624,1,10,52,189,520,1144,
%U A294207 1768,1768,1,11,63,252,772,1916,3684,5452,5452
%N 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.
%H A294207 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LatticePath.html">Lattice Path</a>.
%F A294207 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 A294207 The table begins:
%e A294207 n=0: 1;
%e A294207 n=1: 1;
%e A294207 n=2: 1, 1;
%e A294207 n=3: 1, 2,  2;
%e A294207 n=4: 1, 3,  3;
%e A294207 n=5: 1, 4,  7,  7;
%e A294207 n=6: 1, 5, 12, 19,  19;
%e A294207 n=7: 1, 6, 18, 37,  37;
%e A294207 n=8: 1, 7, 25, 62,  99,  99;
%e A294207 n=9: 1, 8, 33, 95, 194, 293, 293.
%t A294207 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 A294207 Table[T[n, k], {n, 0, 15}, {k, 0, Floor[2n/3]}] // Flatten (* _Jean-François Alcover_, Jul 10 2018 *)
%o A294207 (Sage)
%o A294207 T = [[1]]
%o A294207 for n in range(1,15):
%o A294207     T.append([T[-1][0]])
%o A294207     for k in range(1,floor(2*n/3) + 1):
%o A294207         T[-1].append(T[-1][k-1])
%o A294207         if 2*(n-1)>3*k:
%o A294207             T[-1][-1] += T[-2][k]
%Y A294207 Cf. A009766, A293946.
%K A294207 nonn,tabf
%O A294207 0,6
%A A294207 _Danny Rorabaugh_, Oct 24 2017