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Sum T(n,k) of the entries in the k-th last cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
4

%I #13 Aug 21 2021 06:35:59

%S 1,5,1,25,10,1,143,79,17,1,942,634,197,26,1,7074,5462,2129,417,37,1,

%T 59832,51214,23381,5856,786,50,1,563688,523386,269033,80053,13934,

%U 1360,65,1,5858640,5813892,3281206,1111498,232349,29728,2204,82,1

%N Sum T(n,k) of the entries in the k-th last cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A286231/b286231.txt">Rows n = 1..20, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%e T(3,2) = 10 because the sum of the entries in the second last cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 0+0+3+4+1+2 = 10.

%e Triangle T(n,k) begins:

%e 1;

%e 5, 1;

%e 25, 10, 1;

%e 143, 79, 17, 1;

%e 942, 634, 197, 26, 1;

%e 7074, 5462, 2129, 417, 37, 1;

%e 59832, 51214, 23381, 5856, 786, 50, 1;

%e 563688, 523386, 269033, 80053, 13934, 1360, 65, 1;

%e ...

%Y Column k=1 gives A285382.

%Y Main diagonal and first lower diagonal give: A000012, A002522.

%Y Row sums give A000142 * A000217 = A180119.

%Y Cf. A285793, A286232.

%K nonn,tabl

%O 1,2

%A _Alois P. Heinz_, May 04 2017