A285234 Number of entries in the sixth cycles of all permutations of [n].

1, 23, 382, 5780, 86029, 1301673, 20338679, 330737236, 5618265376, 99849949772, 1857170751804, 36135886878072, 734947859916792, 15608257104179952, 345724111468700496, 7977315239656638912, 191516062334747746752, 4778050475554642998144, 123731984754223222096512

Offset: 6

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 = 6..449
• Wikipedia, Permutation

Crossrefs

Column k=6 of A185105.

Programs

Maple

a:= proc(n) option remember; `if`(n<7, [0$6, 1][n+1],
      ((5*n^3-58*n^2+207*n-230)*a(n-1)-(10*n^4-152*n^3
       +835*n^2-1973*n+1690)*a(n-2)+(n-4)*(10*n^4
       -158*n^3+909*n^2-2251*n+2000)*a(n-3)-(5*n^6
       -127*n^5+1330*n^4-7335*n^3+22396*n^2-35717*n
       +23058)*a(n-4)+(n-5)^6*(n-2)*a(n-5))/((n-3)*(n-6)))
    end:
seq(a(n), n=6..25);

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[Function[p, p + Coefficient[ p, x, 0]*j*x^i][b[n-j, i+1]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]]];
a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]][[6]];
Table[a[n], {n, 6, 25}] (* Jean-François Alcover, Jun 01 2018, after Alois P. Heinz *)

Formula

a(n) = A185105(n,6).

a(n) ~ n!*n/64. - Vaclav Kotesovec, Apr 25 2017