A285231 Number of entries in the third cycles of all permutations of [n].

1, 8, 59, 463, 3978, 37566, 388728, 4385592, 53653680, 708126480, 10034314560, 152001161280, 2451821339520, 41964428419200, 759698874547200, 14505012898790400, 291323663566387200, 6140173922952652800, 135515391451776000000, 3125606951427609600000

Offset

3,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 3..449

Wikipedia, Permutation

Formula

a(n) = A185105(n,3) = A159324(n-1)/2.

a(n) ~ n!*n/8. - Vaclav Kotesovec, Apr 25 2017

Maple

a:= proc(n) option remember; `if`(n<4, [0$3, 1][n+1],
      ((2*n^3-7*n^2+3*n+4)*a(n-1)-(n-2)^3*(n+1)*a(n-2))/(n*(n-3)))
    end:
seq(a(n), n=0..25);

Mathematica

a[3] = 1; a[4] = 8;
a[n_] := a[n] = ((2n^3-7n^2+3n+4) a[n-1] - (n-2)^3 (n+1) a[n-2])/(n(n-3));
Table[a[n], {n, 3, 25}] (* Jean-François Alcover, May 30 2018, from Maple *)

Crossrefs

Column k=3 of A185105.

Cf. A159324.

Sequence in context: A190977 A254662 A186362 * A264817 A032365 A250315

Adjacent sequences: A285228 A285229 A285230 * A285232 A285233 A285234

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2017

Status

approved