A227655 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_{i}-p_{i+1}) <= 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 24, 44, 8, 1, 1, 1, 120, 896, 320, 16, 1, 1, 1, 720, 29392, 33904, 2328, 32, 1, 1, 1, 5040, 1413792, 7453320, 1281696, 16936, 64, 1, 1, 1, 40320, 93770800, 2940381648, 1897242448, 48447504, 123208, 128, 1, 1

Offset: 0

Author

Alois P. Heinz, Jul 19 2013

Keywords

• nonn
• tabl

Examples

			A(2,2) = 2^2 = 4:
        (1,2)       (0,1)
       /     \     /     \
  (2,2)       (1,1)       (0,0)
       \     /     \     /
        (2,1)       (1,0)
Square array A(n,k) begins:
  1, 1,  1,     1,        1,            1, ...
  1, 1,  2,     6,       24,          120, ...
  1, 1,  4,    44,      896,        29392, ...
  1, 1,  8,   320,    33904,      7453320, ...
  1, 1, 16,  2328,  1281696,   1897242448, ...
  1, 1, 32, 16936, 48447504, 482913033152, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..18, flattened

Crossrefs

Columns k=0+1, 2-10 give: A000012, A000079, A227665, A227666, A227667, A227668, A227669, A227670, A227671, A227672.

Rows n=0-10 give: A000012, A000142, A227656, A227657, A227658, A227659, A227660, A227661, A227662, A227663, A227664.

Main diagonal gives A227673.

Cf. A262809, A263159, A318191.

Programs

Maple

b:= proc(l) option remember; `if`({l[]}={0}, 1, add(
      `if`(l[i]=0 or i>1 and 1 `if`(k<2, 1, b([n$k])):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

b[l_] := b[l] = If[Union[l] == {0}, 1, Sum[If[l[[i]] == 0 || i>1 && 1 < Abs[l[[i-1]] - l[[i]] + 1] || i l[[i]]-1]]], {i, 1, Length[l]}]]; a[n_, k_] := If[k<2, 1, b[Array[n&, k]]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)