|
%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
|
