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

1,5

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), ... .

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.

Example

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, ...

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 *)

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.

Sequence in context: A138678 A335997 A350297 * A278792 A343807 A340734

Adjacent sequences: A181728 A181729 A181730 * A181732 A181733 A181734

Keyword

nonn,tabl

Author

Manuel Kauers, Nov 16 2010

Extensions

Edited by Alois P. Heinz, Jul 21 2012

Minor edits by Vaclav Kotesovec, Sep 03 2014

Status

approved