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%I #22 Nov 22 2023 15:38:18
%S 1,6,3,36,24,12,240,180,120,60,1800,1440,1080,720,360,15120,12600,
%T 10080,7560,5040,2520,141120,120960,100800,80640,60480,40320,20160,
%U 1451520,1270080,1088640,907200,725760,544320,362880,181440,16329600,14515200,12700800,10886400,9072000,7257600,5443200,3628800,1814400
%N Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.
%C Row sums = A001809.
%C Column for k = 0 is A001286.
%H Alois P. Heinz, <a href="/A202363/b202363.txt">Rows n = 2..142, flattened</a>
%F E.g.f.: x^2/2 * (1/(1-x)^2)* (1/(1-y*x)).
%e T(3,1) = 3 because from the permutations (given in one line notation): (2,3,1), (3,1,2), (3,2,1) we have respectively 3 inversion pairs (1,2), (2,3) and (1,3) which are all separated by 1 element.
%e Triangle T(n,k) begins:
%e 1;
%e 6, 3;
%e 36, 24, 12;
%e 240, 180, 120, 60;
%e 1800, 1440, 1080, 720, 360;
%e 15120, 12600, 10080, 7560, 5040, 2520;
%e 141120, 120960, 100800, 80640, 60480, 40320, 20160;
%e ...
%t nn=10;Range[0,nn]!CoefficientList[Series[x^2/2/(1-x)^2/(1-y x),{x,0,nn}],{x,y}]//Grid
%Y Cf. A001286, A001809, A055303.
%K nonn,tabl
%O 2,2
%A _Geoffrey Critzer_, Jan 09 2013
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