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A328427 Number of 2n-step walks on cubic lattice starting at (0,0,0), ending at (n,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 A328427 #11 May 13 2020 07:02:48
%S A328427 1,6,93,2040,52086,1443898,42073956,1266666928,39005460010,
%T A328427 1220921798726,38687295919777,1237507863154364,39880747868561408,
%U A328427 1292960017095690800,42125333404129589074,1378096053159532505284,45239194620411006084462,1489461895371375668384236
%N A328427 Number of 2n-step walks on cubic lattice starting at (0,0,0), ending at (n,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 A328427 Alois P. Heinz, <a href="/A328427/b328427.txt">Table of n, a(n) for n = 0..500</a>
%F A328427 a(n) = A328297(2n,n).
%p A328427 b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
%p A328427       add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
%p A328427       sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
%p A328427     end:
%p A328427 a:= n-> add(b(sort([n, j, n-j])), j=0..n):
%p A328427 seq(a(n), n=0..23);
%t A328427 b[l_] := b[l] = If[Last[l] == 0, 1, Function[r, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, r}, {j, r}, {k, r}]][{-1, 0, 1}]];
%t A328427 a[n_] := Sum[b[Sort[{n, j, n - j}]], {j, 0, n}];
%t A328427 a /@ Range[0, 23] (* _Jean-François Alcover_, May 13 2020, after Maple *)
%Y A328427 Cf. A328297.
%K A328427 nonn,walk
%O A328427 0,2
%A A328427 _Alois P. Heinz_, Oct 15 2019