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A098157
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Triangle T(n,k) with diagonals T(n,n-k) = binomial(n+1,2k).
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1
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1, 1, 1, 0, 3, 1, 0, 1, 6, 1, 0, 0, 5, 10, 1, 0, 0, 1, 15, 15, 1, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0, 1, 66, 495, 924, 495, 66, 1, 0, 0, 0, 0, 0, 0, 13, 286, 1287, 1716, 715, 78, 1
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OFFSET
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0,5
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COMMENTS
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Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with k descents. A descent in a permutation a(1)a(2)...a(n) is position i such that a(i)>a(i+1). - Tian Han, Nov 16 2023
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LINKS
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FORMULA
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T(n, k) = binomial(n+1, 2(n-k)) with 0 <= k <= n.
G.f.: (1 + x - q*x)/(1 - 2*q*x - q*x^2 + q^2*x^2). - Tian Han, Nov 16 2023
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EXAMPLE
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Rows begin:
{1},
{1,1},
{0,3,1},
{0,1,6,1),
{0,0,5,10,1},
{0,0,1,15,15,1},
...
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MATHEMATICA
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Table[Binomial[n+1, 2(n-k)], {n, 0, 11}, {k, 0, n}]//Flatten (* Stefano Spezia, Nov 16 2023 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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