A340540 Number of walks of length 4n in the first octant using steps (1,1,1), (-1,0,0), (0,-1,0), and (0,0,-1) that start and end at the origin.

1, 6, 288, 24444, 2738592, 361998432, 53414223552, 8525232846072, 1443209364298944, 255769050813120576, 47020653859202576640, 8907614785269428079168, 1730208409741026141405696, 343266632435192859791576064, 69350551439109880798294334208

Offset: 0

Author

Daniel Carter, Jan 10 2021

Keywords

• nonn
• walk

Comments

There are no such walks with length that is not a multiple of 4.

a(n) is also the number of arrangements of n copies each of "a", "b", "c", and "d" such that no prefix has more b's, c's, or d's than a's.

The analogous problem in dimensions 1 and 2 are given respectively by A000108 (the Catalan numbers) and A006335.

No closed form is known. In fact, it is not known whether this sequence is D-finite (see Bacher et al.).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150
• Axel Bacher, Manuel Kauers, and Rika Yatchak, Continued Classification of 3D Lattice Walks in the Positive Octant, arXiv:1511.05763 [math.CO], 2015.

Crossrefs

Cf. A000108, A006335, A149424.

Column k=3 of A340591.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, 1, `if`(add(i, i=l)+3 x+1, l)), 0) +add(`if`(l[i]>0,
      b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..3))
    end:
a:= n-> b(4*n, [0$3]):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 12 2021

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, 1, If[Total[l] + 3 < n,
  b[n-1, l+1]], 0] + Sum[If[l[[i]] > 0,
  b[n-1, Sort[ReplacePart[l, i -> l[[i]]-1]]], 0], {i, 1, 3}] /. Null -> 0;
a[n_] := b[4n, {0, 0, 0}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 10 2021, after Alois P. Heinz *)

Python

import itertools as it
i = 0
while 1:
    counts = {(a,b,c):0 for a,b,c in it.product(range(i+1), repeat=3)}
    counts[0,0,0] = 1
    for _ in range(4*i):
        update = {(a,b,c):0 for a,b,c in it.product(range(i+1), repeat=3)}
        for x,y,z in counts:
            if counts[x,y,z] != 0:
                for coord in [(x+1,y+1,z+1), (x-1,y,z), (x,y-1,z), (x,y,z-1)]:
                    if coord in update:
                        update[coord] += counts[x,y,z]
        counts = update
    print(i, counts[0,0,0])
    i += 1