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 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). 14
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified March 26 22:49 EDT 2023. Contains 361553 sequences. (Running on oeis4.)