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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

FORMULA

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

EXAMPLE

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;

  ...

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-Fran├žois Alcover, May 10 2020, after Maple *)

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.

Sequence in context: A063394 A193871 A108470 * A157152 A136126 A046802

Adjacent sequences:  A328297 A328298 A328299 * A328301 A328302 A328303

KEYWORD

nonn,tabl,walk

AUTHOR

Alois P. Heinz, Oct 11 2019

STATUS

approved

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Last modified May 14 16:48 EDT 2021. Contains 343898 sequences. (Running on oeis4.)