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, 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

Author

Roger L. Bagula, Jun 13 2008

Keywords

• nonn
• tabf

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

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened
• Mathilde Bouvel, Luca Ferrari, and Bridget Eileen Tenner, Between weak and Bruhat: the middle order on permutations, arXiv:2405.08943 [math.CO], 2024.

Crossrefs

Cf. A000142, A008302 (Mahonian numbers), A006472, A010551, A161680, A259459.

Programs

Mathematica

a := Table[CoefficientList[Product[Sum[D[x^i, x], {i, 1, m}], {m, 1, n}], x], {n, 0, 7}]; Flatten[a]

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)

Extensions

Edited by Alois P. Heinz, May 24 2024