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).

1, 6, 93, 2040, 52086, 1443898, 42073956, 1266666928, 39005460010, 1220921798726, 38687295919777, 1237507863154364, 39880747868561408, 1292960017095690800, 42125333404129589074, 1378096053159532505284, 45239194620411006084462, 1489461895371375668384236

Offset: 0

Author

Alois P. Heinz, Oct 15 2019

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A328297.

Programs

Maple

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
      add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
      sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
    end:
a:= n-> add(b(sort([n, j, n-j])), j=0..n):
seq(a(n), n=0..23);

Mathematica

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}]];
a[n_] := Sum[b[Sort[{n, j, n - j}]], {j, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, May 13 2020, after Maple *)

Formula

a(n) = A328297(2n,n).