A181731 Table A(d,n) of the number of paths of a chess rook in a d-dimensional hypercube from (0...0) to (n...n) where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).

1, 1, 1, 1, 2, 2, 1, 6, 14, 4, 1, 24, 222, 106, 8, 1, 120, 6384, 9918, 838, 16, 1, 720, 291720, 2306904, 486924, 6802, 32, 1, 5040, 19445040, 1085674320, 964948464, 25267236, 56190, 64, 1, 40320, 1781750880, 906140159280, 4927561419120, 439331916888, 1359631776, 470010, 128, 1, 362880, 214899027840, 1224777388630320, 54259623434853360

Offset: 1

Author

Manuel Kauers, Nov 16 2010

Keywords

• nonn
• tabl

Comments

The table is enumerated along antidiagonals: A(1,0), A(2,0), A(1,1), A(3,0), A(2,1), A(1,2), A(4,0), A(3,1), A(2,2), A(1,3), ... .

Examples

			A(3,1) = 6 because there are 6 rook paths on 3D chessboards from (0,0,0) to (1,1,1).
Square table A(d,n) begins:
  1,   1,      2,          4,             8, ...
  1,   2,     14,        106,           838, ...
  1,   6,    222,       9918,        486924, ...
  1,  24,   6384,    2306904,     964948464, ...
  1, 120, 291720, 1085674320, 4927561419120, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..20
• M. Kauers and D. Zeilberger, The Computational Challenge of Enumerating High-Dimensional Rook Walks, arXiv:1011.4671 [math.CO], 2010.

Crossrefs

Rows d=1-12 give: A011782, A051708 (from [1,1]), A144045 (from [1,1,1]), A181749, A181750, A181751, A181752, A181724, A181725, A181726, A181727, A181728.

Columns n=0-2 give: A000012, A000142, A105749.

Main diagonal gives A246623.

Programs

Maple

b:= proc(l) option remember; `if`({l[]} minus {0}={}, 1, add(add
       (b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..nops(l)))
    end:
A:= (d, n)-> b([n$d]):
seq(seq(A(h-n, n), n=0..h-1), h=1..10); # Alois P. Heinz, Jul 21 2012

Mathematica

b[l_List] := b[l] = If[Union[l] ~Complement~ {0} == {}, 1, Sum[ Sum[ b[ Sort[ ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, Length[l]}]]; A[d_, n_] := b[Array[n&, d]]; Table[Table[A[h-n, n], {n, 0, h-1}], {h, 1, 10}] // Flatten (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

Extensions

Edited by Alois P. Heinz, Jul 21 2012

Minor edits by Vaclav Kotesovec, Sep 03 2014