A328297 Number T(n,k) of n-step walks on cubic lattice starting at (0,0,0), ending at (x,y,z) with x=k, remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 2, 1, 5, 6, 1, 16, 26, 14, 1, 58, 112, 93, 30, 1, 228, 489, 522, 288, 62, 1, 945, 2182, 2737, 2040, 825, 126, 1, 4072, 9934, 13934, 12642, 7210, 2254, 254, 1, 18078, 46016, 70058, 72994, 52086, 23878, 5969, 510, 1, 82172, 216322, 350648, 404788, 338520, 198795, 75570, 15468, 1022, 1

Offset: 0

Author

Alois P. Heinz, Oct 11 2019

Keywords

• nonn
• tabl
• walk

Examples

			Triangle T(n,k) begins:
     1;
     2,    1;
     5,    6,     1;
    16,   26,    14,     1;
    58,  112,    93,    30,    1;
   228,  489,   522,   288,   62,    1;
   945, 2182,  2737,  2040,  825,  126,   1;
  4072, 9934, 13934, 12642, 7210, 2254, 254, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Lattice path
• Wikipedia, Self-avoiding walk

Crossrefs

Column k=0 gives A328296.

Main diagonal gives A000012.

T(n,n-1) gives A000918(n+1).

T(2n,n) gives A328427.

Row sums give A328295.

Cf. A038207, A328299, A328300.

Programs

Maple

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
      add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
      sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
    end:
T:= (n, k)-> add(b(sort([k, j, n-k-j])), j=0..n-k):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
T[n_, k_] := Sum[b[Sort[{k, j, n - k - j}]], {j, 0, n - k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 12 2020, after Maple *)