A340590 Number of n*(n+1)-step n-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1.

1, 1, 16, 24444, 8204167296, 1052109889288796160, 78607706455594117933558272000, 4825997038234002956322487606996722432307200, 325844502690869718672482402463320899403011435565608069632000, 31176247959648026790291638390172796940342899651173947284143811081979726010777600

Offset: 0

Author

Alois P. Heinz, Jan 12 2021

Keywords

• nonn
• walk

Examples

			a(2) = 16:
  [(0,0),(1,1),(0,1),(0,0),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(0,1),(0,0),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(0,1),(1,2),(0,2),(0,1),(0,0)],
  [(0,0),(1,1),(0,1),(1,2),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(0,1),(1,2),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(1,0),(0,0),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(1,0),(0,0),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(1,0),(2,1),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(1,0),(2,1),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(1,0),(2,1),(2,0),(1,0),(0,0)],
  [(0,0),(1,1),(2,2),(1,2),(0,2),(0,1),(0,0)],
  [(0,0),(1,1),(2,2),(1,2),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(2,2),(1,2),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(2,2),(2,1),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(2,2),(2,1),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(2,2),(2,1),(2,0),(1,0),(0,0)].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..12

Crossrefs

Main diagonal of A340591.

Cf. A002378, A034841.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, 1, (k-> add(
     `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..k)+
     `if`(add(i, i=l)+k x+1, l)), 0))(nops(l)))
    end:
a:= n-> b(n*(n+1), [0$n]):
seq(a(n), n=0..9);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, 1, Function[k, Sum[
    If[l[[i]]>0, b[n-1, Sort[ReplacePart[l, i -> l[[i]]-1]]], 0], {i, 1, k}] +
    If[Sum[i, {i, l}] + k < n, b[n - 1, Map[#+1&, l]], 0]][Length[l]]];
a[n_] := b[n(n+1), Table[0, {n}]];
a /@ Range[0, 9] (* Jean-François Alcover, Jan 26 2021, after Alois P. Heinz *)

Formula

a(n) = A340591(n,n).