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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

%I #26 May 10 2020 19:35:11

%S 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,

%T 1,1,127,651,1257,1257,651,127,1,1,255,1730,4256,5594,4256,1730,255,1,

%U 1,511,4494,13669,22411,22411,13669,4494,511,1,1,1023,11485,42279,83680,103730,83680,42279,11485,1023,1

%N 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.

%H Alois P. Heinz, <a href="/A328300/b328300.txt">Rows n = 0..200, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>

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

%e Triangle T(n,k) begins:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 7, 7, 1;

%e 1, 15, 26, 15, 1;

%e 1, 31, 82, 82, 31, 1;

%e 1, 63, 237, 343, 237, 63, 1;

%e 1, 127, 651, 1257, 1257, 651, 127, 1;

%e 1, 255, 1730, 4256, 5594, 4256, 1730, 255, 1;

%e ...

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

%p add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(

%p sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))

%p end:

%p T:= (n, k)-> b(sort([0, k, n-k])):

%p seq(seq(T(n, k), k=0..n), n=0..12);

%t 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 T[n_, k_] := b[Sort[{0, k, n - k}]];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 10 2020, after Maple *)

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

%Y Row sums give A328296.

%Y T(2n,n) gives A328269.

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

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

%K nonn,tabl,walk

%O 0,5

%A _Alois P. Heinz_, Oct 11 2019

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Last modified March 29 05:43 EDT 2024. Contains 371264 sequences. (Running on oeis4.)