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A139769
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T(n,k) = [x^k] Product_{m=1..n} d/dx Sum_{i=1..m} x^i; triangle read by rows, n >= 0, 0 <= k <= A161680(n).
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1
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1, 1, 1, 2, 1, 4, 7, 6, 1, 6, 18, 36, 49, 46, 24, 1, 8, 33, 94, 204, 354, 497, 562, 501, 326, 120, 1, 10, 52, 188, 528, 1222, 2406, 4102, 6116, 7996, 9132, 9014, 7541, 5116, 2556, 720, 1, 12, 75, 326, 1105, 3106, 7513, 16014, 30558, 52752, 82938, 119230, 156983
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OFFSET
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0,4
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COMMENTS
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T(n, binomial(n,2)-k) is the number of rank-k intervals in the middle order on permutations. (See Bouvel et al. reference.) - Bridget Tenner, May 24 2024
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LINKS
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FORMULA
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|Sum_{k=0..binomial(n,2)} (-1)^k T(n,k)| = A010551(n).
Sum_{k=0..binomial(n,2)} (binomial(n,2)-k)*T(n,k) = A259459(n-2) for n>=2. (End)
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EXAMPLE
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Triangle T(n,k) begins:
1;
1;
1, 2;
1, 4, 7, 6;
1, 6, 18, 36, 49, 46, 24;
1, 8, 33, 94, 204, 354, 497, 562, 501, 326, 120;
...
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MATHEMATICA
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a := Table[CoefficientList[Product[Sum[D[x^i, x], {i, 1, m}], {m, 1, n}], x], {n, 0, 7}]; Flatten[a]
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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