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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).

%I A139769 #31 May 25 2024 12:11:18
%S A139769 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,
%T A139769 120,1,10,52,188,528,1222,2406,4102,6116,7996,9132,9014,7541,5116,
%U A139769 2556,720,1,12,75,326,1105,3106,7513,16014,30558,52752,82938,119230,156983
%N 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).
%C A139769 Row sums are A006472(n+1).
%C A139769 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
%H A139769 Alois P. Heinz, <a href="/A139769/b139769.txt">Rows n = 0..50, flattened</a>
%H A139769 Mathilde Bouvel, Luca Ferrari, and Bridget Eileen Tenner, <a href="https://arxiv.org/abs/2405.08943">Between weak and Bruhat: the middle order on permutations</a>, arXiv:2405.08943 [math.CO], 2024.
%F A139769 From _Alois P. Heinz_, May 24 2024: (Start)
%F A139769 |Sum_{k=0..binomial(n,2)} (-1)^k T(n,k)| = A010551(n).
%F A139769 Sum_{k=0..binomial(n,2)} (binomial(n,2)-k)*T(n,k) = A259459(n-2) for n>=2. (End)
%e A139769 Triangle T(n,k) begins:
%e A139769   1;
%e A139769   1;
%e A139769   1, 2;
%e A139769   1, 4,  7,  6;
%e A139769   1, 6, 18, 36,  49,  46,  24;
%e A139769   1, 8, 33, 94, 204, 354, 497, 562, 501, 326, 120;
%e A139769   ...
%t A139769 a := Table[CoefficientList[Product[Sum[D[x^i, x], {i, 1, m}], {m, 1, n}], x], {n, 0, 7}]; Flatten[a]
%Y A139769 Cf. A000142, A008302 (Mahonian numbers), A006472, A010551, A161680, A259459.
%K A139769 nonn,tabf
%O A139769 0,4
%A A139769 _Roger L. Bagula_, Jun 13 2008
%E A139769 Edited by _Alois P. Heinz_, May 24 2024