A328300 Number T(n,k) of n-step walks on cubic lattice starting at (0,0,0), ending at (0,k,n-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, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 26, 15, 1, 1, 31, 82, 82, 31, 1, 1, 63, 237, 343, 237, 63, 1, 1, 127, 651, 1257, 1257, 651, 127, 1, 1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1, 1, 511, 4494, 13669, 22411, 22411, 13669, 4494, 511, 1, 1, 1023, 11485, 42279, 83680, 103730, 83680, 42279, 11485, 1023, 1

Offset: 0

Author

Alois P. Heinz, Oct 11 2019

Keywords

• nonn
• tabl
• walk

Examples

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,    1;
  1,  15,   26,   15,    1;
  1,  31,   82,   82,   31,    1;
  1,  63,  237,  343,  237,   63,    1;
  1, 127,  651, 1257, 1257,  651,  127,   1;
  1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Wikipedia, Lattice path
• Wikipedia, Self-avoiding walk

Crossrefs

Columns k=0-1 give: A000012, A000225.

Row sums give A328296.

T(2n,n) gives A328269.

T(n,floor(n/2)) gives A328280.

Cf. A007318, A008288, A091533, A328297, A328347.

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)-> b(sort([0, k, n-k])):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[l_List] := b[l] = If[l[[-1]] == 0, 1, Function[r, Sum[Sum[Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {k, r}], {j, r}], {i, r}]][{-1, 0, 1}]];
T[n_, k_] := b[Sort[{0, k, n - k}]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 10 2020, after Maple *)

Formula

T(n,k) = T(n,n-k).