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A214776
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Number A(n,k) of standard Young tableaux of shape [n*k,n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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19
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1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 9, 5, 0, 1, 4, 20, 48, 14, 0, 1, 5, 35, 154, 275, 42, 0, 1, 6, 54, 350, 1260, 1638, 132, 0, 1, 7, 77, 663, 3705, 10659, 9996, 429, 0, 1, 8, 104, 1120, 8602, 40480, 92092, 62016, 1430, 0, 1, 9, 135, 1748, 17199, 115101, 451269, 807300, 389367, 4862, 0
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OFFSET
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0,8
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COMMENTS
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A(n,k) is also the number of binary words with n*k 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The A(2,2) = 9 words are: 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100.
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LINKS
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FORMULA
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A(n,k) = max(0, C((k+1)*n,n)*((k-1)*n+1)/(k*n+1)).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 9, 20, 35, 54, 77, ...
0, 5, 48, 154, 350, 663, 1120, ...
0, 14, 275, 1260, 3705, 8602, 17199, ...
0, 42, 1638, 10659, 40480, 115101, 272272, ...
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MAPLE
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A:= (n, k)-> max(0, binomial((k+1)*n, n)*((k-1)*n+1)/(k*n+1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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a[n_, k_] := Max[0, Binomial[(k+1)*n, n]*((k-1)*n+1)/(k*n+1)]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 01 2013, after Maple *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000108, A174687, A126596, A215541, A215542, A215551, A215552, A215553, A215554, A215555.
Rows n=0-10 give: A000012, A001477, A014107(k+1), A215543, A215544, A215545, A215546, A215547, A215548, A215549, A215550.
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KEYWORD
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AUTHOR
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STATUS
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approved
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