A328347 - OEIS

%I #25 Apr 30 2020 07:13:49

%S 1,1,1,3,4,3,7,15,15,7,19,52,72,52,19,51,175,300,300,175,51,141,576,

%T 1185,1480,1185,576,141,393,1869,4473,6685,6685,4473,1869,393,1107,

%U 6000,16380,28392,33880,28392,16380,6000,1107,3139,19107,58572,115332,159264,159264,115332,58572,19107,3139

%N 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.

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

%H Alois P. Heinz, <a href="/A328347/b328347.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      3,    4,     3;

%e      7,   15,    15,     7;

%e     19,   52,    72,    52,    19;

%e     51,  175,   300,   300,   175,    51;

%e    141,  576,  1185,  1480,  1185,   576,   141;

%e    393, 1869,  4473,  6685,  6685,  4473,  1869,  393;

%e   1107, 6000, 16380, 28392, 33880, 28392, 16380, 6000, 1107;

%e   ...

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

%p       add(`if`(i+j+k=1, (h-> `if`(add(t, t=h)<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, 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 T[n_, k_] := b[Sort[{0, k, n - k}]];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)

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

%Y Row sums give A084609.

%Y T(2n,n) gives A328426.

%Y Cf. A007318, A328300, A328345.

%K nonn,tabl,walk

%O 0,4

%A _Alois P. Heinz_, Oct 13 2019