A285233 Number of entries in the fifth cycles of all permutations of [n].

1, 17, 221, 2724, 34009, 441383, 6020276, 86673088, 1318681308, 21194234508, 359421505224, 6421154849208, 120637782989568, 2379195625677696, 49167226489281408, 1062833010282628992, 23992442301958329600, 564697104190192569600, 13836823816466433139200

Offset: 5

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 = 5..449
• Wikipedia, Permutation

Crossrefs

Column k=5 of A185105.

Programs

Maple

a:= proc(n) option remember; `if`(n<6, [0$5, 1][n+1],
      ((4*(n^3-9*n^2+24*n-19))*a(n-1)-(6*n^4-72*n^3+
       307*n^2-547*n+334)*a(n-2)+(4*n^5-64*n^4+398*n^3
      -1191*n^2+1683*n-862)*a(n-3)-(n-4)^5*(n-1)*a(n-4))
      /((n-2)*(n-5)))
    end:
seq(a(n), n=5..25);

Mathematica

a[3] = a[4] = 0; a[5] = 1; a[6] = 17; a[n_] := a[n] = ((4(n^3 - 9n^2 + 24n - 19)) a[n-1] - (6n^4 - 72n^3 + 307n^2 - 547n + 334) a[n-2] + (4n^5 - 64n^4 + 398n^3 - 1191n^2 + 1683n - 862) a[n-3] - (n-4)^5 (n-1) a[n-4]) / ((n - 2)(n - 5));
Table[a[n], {n, 5, 25}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

Formula

a(n) = A185105(n,5).

a(n) ~ n!*n/32. - Vaclav Kotesovec, Apr 25 2017