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A328347 Number T(n,k) of n-step walks on cubic lattice starting at (0,0,0), ending at (0,k,n-k) 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. 7
1, 1, 1, 3, 4, 3, 7, 15, 15, 7, 19, 52, 72, 52, 19, 51, 175, 300, 300, 175, 51, 141, 576, 1185, 1480, 1185, 576, 141, 393, 1869, 4473, 6685, 6685, 4473, 1869, 393, 1107, 6000, 16380, 28392, 33880, 28392, 16380, 6000, 1107, 3139, 19107, 58572, 115332, 159264, 159264, 115332, 58572, 19107, 3139 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

These walks are not restricted to the first (nonnegative) octant.

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;

     3,    4,     3;

     7,   15,    15,     7;

    19,   52,    72,    52,    19;

    51,  175,   300,   300,   175,    51;

   141,  576,  1185,  1480,  1185,   576,   141;

   393, 1869,  4473,  6685,  6685,  4473,  1869,  393;

  1107, 6000, 16380, 28392, 33880, 28392, 16380, 6000, 1107;

  ...

MAPLE

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(add(

      add(`if`(i+j+k=1, (h-> `if`(add(t, t=h)<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, Sum[If[i + j + k == 1, Function[h, If[Total[h] < 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_] := b[Sort[{0, k, n - k}]];

Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Apr 30 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=0-1 give: A002426, A132894 = n*A005773(n).

Row sums give A084609.

T(2n,n) gives A328426.

Cf. A007318, A328300, A328345.

Sequence in context: A333974 A163108 A077005 * A265723 A134065 A015887

Adjacent sequences:  A328344 A328345 A328346 * A328348 A328349 A328350

KEYWORD

nonn,tabl,walk

AUTHOR

Alois P. Heinz, Oct 13 2019

STATUS

approved

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Last modified July 27 02:39 EDT 2021. Contains 346302 sequences. (Running on oeis4.)