A285232 Number of entries in the fourth cycles of all permutations of [n].

1, 12, 119, 1177, 12217, 135302, 1606446, 20450052, 278604252, 4051377504, 62702222112, 1029832270848, 17899305402240, 328353314855040, 6341705227082880, 128655706986282240, 2735782096305749760, 60855815849067648000, 1413487524282196608000

Offset: 4

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 = 4..449
• Wikipedia, Permutation

Crossrefs

Column k=4 of A185105.

Programs

Maple

a:= proc(n) option remember; `if`(n<5, [0$4, 1][n+1],
      ((n-2)*(3*n^2-13*n+6)*a(n-1)-(3*n^4-26*n^3+76*n^2
      -81*n+16)*a(n-2)+(n-3)^4*n*a(n-3))/((n-1)*(n-4)))
    end:
seq(a(n), n=4..30);

Mathematica

a[3]=0; a[4]=1; a[5]=12; a[n_] := a[n] = ((n-2)(3n^2 - 13n + 6) a[n-1] - ( 3n^4 - 26n^3 + 76n^2 - 81n + 16)a[n-2]+(n-3)^4 n a[n-3])/((n-1)(n-4));
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

Formula

a(n) = A185105(n,4).

a(n) ~ n!*n/16. - Vaclav Kotesovec, Apr 25 2017