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A181731
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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).
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14
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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
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OFFSET
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1,5
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COMMENTS
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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), ... .
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LINKS
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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.
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EXAMPLE
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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, ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn,tabl
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AUTHOR
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Manuel Kauers, Nov 16 2010
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EXTENSIONS
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Edited by Alois P. Heinz, Jul 21 2012
Minor edits by Vaclav Kotesovec, Sep 03 2014
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STATUS
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approved
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