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

Author

Alois P. Heinz, Jan 26 2021

Keywords

• nonn
• walk

Examples

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

Links

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

Crossrefs

Main diagonal of A335570.

Programs

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 *)

Formula

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

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