A335588 Number of n-step n-dimensional nonnegative lattice walks starting at the origin and using steps that increment all components or decrement one component by 1.

1, 1, 3, 13, 81, 686, 7525, 102173, 1655241, 31119382, 665254791, 15927737772, 422179410829, 12275253219828, 388591800808471, 13309116622983421, 490515662121994785, 19362705183912628838, 815258217524407553989, 36479395828632610279316, 1729012534789121191076601

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..65 (terms 0..55 from Alois P. Heinz)

Formula

a(n) = A335570(n,n).

a(n) == 1 (mod n) for n >= 2.

Example

a(2) = 3: [(0,0),(1,1),(2,2)], [(0,0),(1,1),(0,1)], [(0,0),(1,1),(1,0)].

Maple

b:= proc(n, l) option remember; `if`(n=0, 1, b(n-1, map(x-> x+1, l))+add(
     `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..nops(l)))
    end:
a:= n-> b(n, [0$n]):
seq(a(n), n=0..23);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, 1, b[n - 1, l + 1] + Sum[If[l[[i]] > 0, b[n - 1, Sort[ReplacePart[l, i -> l[[i]] - 1]]], 0], {i, 1, Length[l]}]];
a[n_] := b[n, Table[0, {n}]];
a /@ Range[0, 23] (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)

Crossrefs

Main diagonal of A335570.

Sequence in context: A160882 A135921 A005923 * A089461 A000684 A222272

Adjacent sequences: A335585 A335586 A335587 * A335589 A335590 A335591

Keyword

nonn,walk

Author

Alois P. Heinz, Jan 26 2021

Status

approved