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A088459
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Triangle read by rows: T(n,k) represents the number of lozenge tilings of an (n,1,n)-hexagon which include the non-vertical tile above the main diagonal starting in position k+1.
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16
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1, 1, 1, 2, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 7, 42, 126, 315, 525, 700, 700, 525, 315, 126, 42, 7, 1, 1, 8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1
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OFFSET
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1,4
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COMMENTS
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Rows are of length 2, 4, 6, 8, 10, 12, ...
T(n,k)= number of symmetric Dyck paths of length 4n and having k peaks. Example: T(2,3)=2 because we have UU*DU*DU*DD and U*DUU*DDU*D, where U=(1,1), D=(1,-1) and * shows the peaks. - Emeric Deutsch, Feb 22 2004
T(n,k) is also the number of nodes at distance k from a specified node in the n-odd graph for k in 1..n-1. - Eric W. Weisstein, Mar 23 2018
T(n,k) seems to be the k-th Lie-Betti number of the star graph on n vertices. See A360571 for additional information and references. - Samuel J. Bevins, Feb 12 2023
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LINKS
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FORMULA
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T(n, k) = binomial(n, ceiling(k/2))* binomial(n-1, floor(k/2)), n>0 and k=0 to 2n-1.
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EXAMPLE
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For example, the number of tilings of a 4,1,4 hexagon which includes the non-vertical tile above the main diagonal starting in position 3 is T(4,2)=12.
Triangle T(n, k) begins:
[1] 1,1,
[2] 1,2, 2, 1,
[3] 1,3, 6, 6, 3, 1,
[4] 1,4,12, 18, 18, 12, 4, 1,
[5] 1,5,20, 40, 60, 60, 40, 20, 5, 1,
[6] 1,6,30, 75, 150, 200, 200, 150, 75, 30, 6, 1,
[7] 1,7,42,126, 315, 525, 700, 700, 525, 315, 126, 42, 7, 1,
[8] 1,8,56,196, 588,1176,1960,2450,2450,1960,1176,588, 196, 56, 8, 1,
[9] 1,9,72,288,1008,2352,4704,7056,8820,8820,7056,4704,2352,1008,288,72,9,1
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MAPLE
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binomial(n, ceil(k/2))*binomial(n-1, floor(k/2)) ;
end proc:
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MATHEMATICA
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Table[Binomial[n, Ceiling[k/2]] Binomial[n - 1, Floor[k/2]], {n, 10}, {k, 0, 2 n - 1}] // Flatten (* Eric W. Weisstein, Mar 23 2018 *)
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CROSSREFS
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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Christopher Hanusa (chanusa(AT)washington.edu), Nov 14 2003
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EXTENSIONS
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STATUS
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approved
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