A340591 Number A(n,k) of n*(k+1)-step k-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 16, 5, 1, 1, 24, 288, 192, 14, 1, 1, 120, 9216, 24444, 2816, 42, 1, 1, 720, 460800, 7303104, 2738592, 46592, 132, 1, 1, 5040, 33177600, 4234233600, 8204167296, 361998432, 835584, 429, 1, 1, 40320, 3251404800, 4223111040000, 59027412643200, 11332298092032, 53414223552, 15876096, 1430, 1

Offset: 0

Author

Alois P. Heinz, Jan 12 2021

Keywords

• nonn
• tabl
• walk

Examples

			Square array A(n,k) begins:
  1,  1,     1,         1,              1,                   1, ...
  1,  1,     2,         6,             24,                 120, ...
  1,  2,    16,       288,           9216,              460800, ...
  1,  5,   192,     24444,        7303104,          4234233600, ...
  1, 14,  2816,   2738592,     8204167296,      59027412643200, ...
  1, 42, 46592, 361998432, 11332298092032, 1052109889288796160, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..24, flattened

Crossrefs

Columns k=0-3 give: A000012, A000108, A006335, A340540.

Rows n=0-2 give: A000012, A000142, |A055546|.

Main diagonal gives A340590.

Cf. A335570.

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, k)-> b(k*n+n, [0$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);

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_, k_] := b[k*n + n, Table[0, {k}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 26 2021, after Alois P. Heinz *)