A328426 Number of 2n-step walks on cubic lattice starting at (0,0,0), ending at (0,n,n) and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

1, 4, 72, 1480, 33880, 819504, 20562696, 529399728, 13892650200, 369988864960, 9969694635472, 271217782469088, 7436910593958664, 205289086309888000, 5699233814774374800, 159004177074207384480, 4455230052707022818520, 125307788218001246547360

Offset

0,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Formula

a(n) = A328347(2n,n).

Maple

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(add(add(
      `if`(i+j+k=1, (h-> `if`((w->w<0 or w<max(map(abs, h)))(add(t, t=h)),
       0, b(h)))(sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
    end:
a:= n-> b(sort([0, n$2])):
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[Function[w, w < 0 || w < Max[Map[Abs, h]]][Total[h]], 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, r}, {j, r}, {k, r}]][{-1, 0, 1}]];
a[n_] := b[Sort[{0, n, n}]];
a /@ Range[0, 23] (* Jean-François Alcover, May 13 2020, after Maple *)

Crossrefs

Cf. A328347.

Sequence in context: A231033 A307358 A201976 * A176901 A304316 A186415

Adjacent sequences: A328423 A328424 A328425 * A328427 A328428 A328429

Keyword

nonn,walk

Author

Alois P. Heinz, Oct 15 2019

Status

approved