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A328296 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (0,y,z), 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).

%I A328296 #16 May 12 2020 11:50:41
%S A328296 1,2,5,16,58,228,945,4072,18078,82172,380666,1791138,8537912,41146988,
%T A328296 200169891,981705400,4848820372,24098703860,120433164750,604831645542,
%U A328296 3050979757728,15451575335362,78536766518038,400497435480332,2048473941706016,10506489209380466
%N A328296 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (0,y,z), 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).
%H A328296 Alois P. Heinz, <a href="/A328296/b328296.txt">Table of n, a(n) for n = 0..610</a>
%H A328296 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A328296 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%F A328296 a(n) is odd <=> n in { A000918 } and n >= 0.
%p A328296 b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
%p A328296       add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
%p A328296       sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
%p A328296     end:
%p A328296 a:= n-> add(b(sort([0, j, n-j])), j=0..n):
%p A328296 seq(a(n), n=0..29);
%t A328296 b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
%t A328296 a[n_] := Sum[b[Sort[{0, j, n  - j}]], {j, 0, n }];
%t A328296 a /@ Range[0, 29] (* _Jean-François Alcover_, May 12 2020, after Maple *)
%Y A328296 Column k=0 of A328297.
%Y A328296 Row sums of A328300.
%Y A328296 Cf. A000079, A000918.
%K A328296 nonn,walk
%O A328296 0,2
%A A328296 _Alois P. Heinz_, Oct 11 2019