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A285793 Sum T(n,k) of the k-th entries in all cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A285793 #14 May 05 2017 10:58:33
%S A285793 1,4,2,18,13,5,96,83,43,18,600,582,342,192,84,4320,4554,2874,1824,
%T A285793 1068,480,35280,39672,26232,17832,11784,7080,3240,322560,382248,
%U A285793 261288,185688,131256,88920,54360,25200,3265920,4044240,2834640,2078640,1534320,1110960,765360,473760,221760
%N A285793 Sum T(n,k) of the k-th entries in all cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%C A285793 Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%H A285793 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A285793 T(n,1) = n * n!.
%F A285793 T(n,n) = floor((n-1)!*(n+2)/2).
%e A285793 T(3,2) = 13 because the sum of the second entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 2+3+2+3+3+0 = 13.
%e A285793 Triangle T(n,k) begins:
%e A285793 :      1;
%e A285793 :      4,      2;
%e A285793 :     18,     13,      5;
%e A285793 :     96,     83,     43,     18;
%e A285793 :    600,    582,    342,    192,     84;
%e A285793 :   4320,   4554,   2874,   1824,   1068,   480;
%e A285793 :  35280,  39672,  26232,  17832,  11784,  7080,  3240;
%e A285793 : 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200;
%Y A285793 Columns k=1-2 give: A001563, A285795.
%Y A285793 Main diagonal and first lower diagonal give: A038720(n-1) (for n>1), A286175.
%Y A285793 Row sums give A000142 * A000217 = A180119.
%Y A285793 Cf. A285439, A285595, A286231.
%K A285793 nonn,tabl
%O A285793 1,2
%A A285793 _Alois P. Heinz_, Apr 26 2017