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

0,2

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

Example

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;
  ...

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 *)

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.

Sequence in context: A217204 A179455 A039810 * A124575 A178121 A302595

Adjacent sequences: A328294 A328295 A328296 * A328298 A328299 A328300

Keyword

nonn,tabl,walk

Author

Alois P. Heinz, Oct 11 2019

Status

approved