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.

1, 4, 2, 18, 13, 5, 96, 83, 43, 18, 600, 582, 342, 192, 84, 4320, 4554, 2874, 1824, 1068, 480, 35280, 39672, 26232, 17832, 11784, 7080, 3240, 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200, 3265920, 4044240, 2834640, 2078640, 1534320, 1110960, 765360, 473760, 221760

Offset

1,2

Comments

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Links

Table of n, a(n) for n=1..45.

Wikipedia, Permutation

Formula

T(n,1) = n * n!.

T(n,n) = floor((n-1)!*(n+2)/2).

Example

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.
Triangle T(n,k) begins:
:      1;
:      4,      2;
:     18,     13,      5;
:     96,     83,     43,     18;
:    600,    582,    342,    192,     84;
:   4320,   4554,   2874,   1824,   1068,   480;
:  35280,  39672,  26232,  17832,  11784,  7080,  3240;
: 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200;

Crossrefs

Columns k=1-2 give: A001563, A285795.

Main diagonal and first lower diagonal give: A038720(n-1) (for n>1), A286175.

Row sums give A000142 * A000217 = A180119.

Cf. A285439, A285595, A286231.

Sequence in context: A074676 A257505 A152883 * A117692 A052966 A305135

Adjacent sequences: A285790 A285791 A285792 * A285794 A285795 A285796

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 26 2017

Status

approved