%I #24 Jun 04 2022 21:31:00
%S 1,0,2,0,1,4,1,2,1,0,0,4,8,4,10,10,6,2,0,0,1,12,18,16,35,44,47,40,25,
%T 14,8,4,1,0,0,0,6,32,44,60,118,160,208,244,244,214,174,140,104,64,30,
%U 10,2,0,0,0,1,24,83,118,206,388,565,802,1068,1308,1466,1508,1479,1414,1290,1076,806,544,333,186,96,46,19,6,1
%N Irregular triangular array read by rows: T(n,k) is the number of derangement permutations of [n] that have exactly k inversions; n>=2, 1<=k<=binomial(n,2) for even n, 1<=k<=binomial(n,2)-1 for odd n.
%C Row sums = A000166.
%C Sum_{k>=1} T(n,k)*k = A216239(n).
%C Sum_{even k} T(n,k) = A003221(n) and Sum_{odd k} T(n,k) = A000387(n).
%C It would be nice to have a closed formula for T(n,k). - _Alois P. Heinz_, Dec 31 2014
%H Alois P. Heinz, <a href="/A228924/b228924.txt">Rows n = 2..23, flattened</a>
%e Triangle T(n,k) begins:
%e 1;
%e 0, 2;
%e 0, 1, 4, 1, 2, 1;
%e 0, 0, 4, 8, 4, 10, 10, 6, 2;
%e 0, 0, 1, 12, 18, 16, 35, 44, 47, 40, 25, 14, 8, 4, 1;
%e ...
%t Map[Distribution[#,Range[1,Max[#]]]&,Table[Map[Inversions,Derangements[n]],{n,2,6}]]//Grid
%Y Cf. A000166, A000387, A003221, A216239.
%K nonn,tabf
%O 2,3
%A _Geoffrey Critzer_, Sep 08 2013