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A139769
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).
1
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
OFFSET
0,4
COMMENTS
Row sums are A006472(n+1).
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
LINKS
Mathilde Bouvel, Luca Ferrari, and Bridget Eileen Tenner, Between weak and Bruhat: the middle order on permutations, arXiv:2405.08943 [math.CO], 2024.
FORMULA
From Alois P. Heinz, May 24 2024: (Start)
|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)
EXAMPLE
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;
...
MATHEMATICA
a := Table[CoefficientList[Product[Sum[D[x^i, x], {i, 1, m}], {m, 1, n}], x], {n, 0, 7}]; Flatten[a]
CROSSREFS
Cf. A000142, A008302 (Mahonian numbers), A006472, A010551, A161680, A259459.
Sequence in context: A322941 A217205 A358566 * A357470 A326894 A378343
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Jun 13 2008
EXTENSIONS
Edited by Alois P. Heinz, May 24 2024
STATUS
approved