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

Author

Alois P. Heinz, Apr 15 2017

Keywords

• nonn

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

Crossrefs

Column k=3 of A185105.

Cf. A159324.

Programs

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 *)

Formula

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

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