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%I #32 Jul 24 2019 15:24:22
%S 1,1,1,1,2,2,1,6,14,4,1,24,222,106,8,1,120,6384,9918,838,16,1,720,
%T 291720,2306904,486924,6802,32,1,5040,19445040,1085674320,964948464,
%U 25267236,56190,64,1,40320,1781750880,906140159280,4927561419120,439331916888,1359631776,470010,128,1,362880,214899027840,1224777388630320,54259623434853360
%N 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).
%C 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), ... .
%H Alois P. Heinz, <a href="/A181731/b181731.txt">Antidiagonals n = 1..20</a>
%H M. Kauers and D. Zeilberger, <a href="http://arxiv.org/abs/1011.4671">The Computational Challenge of Enumerating High-Dimensional Rook Walks</a>, arXiv:1011.4671 [math.CO], 2010.
%e A(3,1) = 6 because there are 6 rook paths on 3D chessboards from (0,0,0) to (1,1,1).
%e Square table A(d,n) begins:
%e 1, 1, 2, 4, 8, ...
%e 1, 2, 14, 106, 838, ...
%e 1, 6, 222, 9918, 486924, ...
%e 1, 24, 6384, 2306904, 964948464, ...
%e 1, 120, 291720, 1085674320, 4927561419120, ...
%p b:= proc(l) option remember; `if`({l[]} minus {0}={}, 1, add(add
%p (b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..nops(l)))
%p end:
%p A:= (d, n)-> b([n$d]):
%p seq(seq(A(h-n, n), n=0..h-1), h=1..10); # _Alois P. Heinz_, Jul 21 2012
%t 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_ *)
%Y Rows d=1-12 give: A011782, A051708 (from [1,1]), A144045 (from [1,1,1]), A181749, A181750, A181751, A181752, A181724, A181725, A181726, A181727, A181728.
%Y Columns n=0-2 give: A000012, A000142, A105749.
%Y Main diagonal gives A246623.
%K nonn,tabl
%O 1,5
%A _Manuel Kauers_, Nov 16 2010
%E Edited by _Alois P. Heinz_, Jul 21 2012
%E Minor edits by _Vaclav Kotesovec_, Sep 03 2014
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