A318191 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 we have abs(p_{i}-p_{(i mod k)+1}) <= 1 and the first component used is p_1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 6, 12, 4, 1, 1, 1, 24, 180, 72, 8, 1, 1, 1, 120, 4680, 5400, 432, 16, 1, 1, 1, 720, 187200, 914400, 162000, 2592, 32, 1, 1, 1, 5040, 10634400, 296438400, 178660800, 4860000, 15552, 64, 1, 1, 1, 40320, 813664800, 162273628800, 469551168000, 34907788800, 145800000, 93312, 128, 1, 1

Offset: 0

Author

Alois P. Heinz, Jan 07 2019

Keywords

• nonn
• tabl

Examples

			A(2,2) = 2^2 = 4:
                    (0,1)
                   /     \
  (2,2)-(1,2)-(1,1)       (0,0)
                   \     /
                    (1,0)
Square array A(n,k) begins:
  1, 1,  1,     1,         1,             1,                   1, ...
  1, 1,  1,     2,         6,            24,                 120, ...
  1, 1,  2,    12,       180,          4680,              187200, ...
  1, 1,  4,    72,      5400,        914400,           296438400, ...
  1, 1,  8,   432,    162000,     178660800,        469551168000, ...
  1, 1, 16,  2592,   4860000,   34907788800,     743761386086400, ...
  1, 1, 32, 15552, 145800000, 6820487308800, 1178106009360998400, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..20, flattened

Crossrefs

Columns k=0+1, 2 give: A000012, A011782.

Rows n=0-2 give: A000012, A000142(n-1) for n>0, A322782/n for n>0.

Main diagonal gives A320443.

Cf. A227655.

Programs

Maple

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

Mathematica

b[l_] := b[l] = With[{n = Length[l]}, If[n < 2 || Max[l ] == 0, 1, Sum[If[ l[[i]] == 0 ||1 < Abs[l[[If[i == 1, 0, i] - 1]] - l[[i]] + 1] || 1 < Abs[l[[If[i == n, 0, i] + 1]] - l[[i]] + 1], 0, b[ReplacePart[l, i -> l[[i]] - 1]]], {i, n}]]];
A[n_, k_] :=  If[k < 2 || n == 0, 1, b[Join[{n - 1}, Table[n, {k - 1}]]]];
Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 13 2020, after Maple *)