A285236 Number of entries in the eighth cycles of all permutations of [n].

1, 38, 964, 21018, 431007, 8671656, 175065071, 3591984289, 75473055841, 1631318215818, 36369569578502, 837619857754240, 19943142053389024, 491010028537071248, 12499878460133012064, 328936666440527737296, 8943724877454118086096, 251125623168859020015072

Offset: 8

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 = 8..450
• Wikipedia, Permutation

Crossrefs

Column k=8 of A185105.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add((p-> p+`if`(i=1, coeff(p, x, 0)*j*x, 0))(
      b(n-j, max(0, i-1)))*binomial(n-1, j-1)*
      (j-1)!, j=1..n)))
    end:
a:= n-> coeff(b(n, 8), x, 1):
seq(a(n), n=8..30);

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, Sum[Function[p, p + If[i == 1, Coefficient[p, x, 0]*j*x, 0]][b[n - j, Max[0, i - 1]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]]];
a[n_] := Coefficient[b[n, 8], x, 1];
Table[a[n], {n, 8, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

Formula

a(n) = A185105(n,8).

Recurrence: (n-8)*(n-5)*a(n) = (7*n^3 - 117*n^2 + 618*n - 1036)*a(n-1) - (21*n^4 - 450*n^3 + 3521*n^2 - 11996*n + 15092)*a(n-2) + 5*(7*n^5 - 190*n^4 + 2039*n^3 - 10842*n^2 + 28614*n - 30016)*a(n-3) - (35*n^6 - 1185*n^5 + 16635*n^4 - 124015*n^3 + 518011*n^2 - 1149493*n + 1058400)*a(n-4) + (n-6)*(21*n^6 - 747*n^5 + 11033*n^4 - 86597*n^3 + 380805*n^2 - 888917*n + 859586)*a(n-5) - (7*n^8 - 352*n^7 + 7728*n^6 - 96726*n^5 + 754656*n^4 - 3756732*n^3 + 11646888*n^2 - 20547489*n + 15780868)*a(n-6) + (n-7)^8*(n-4)*a(n-7), for n>8. - Vaclav Kotesovec, Apr 25 2017

a(n) ~ n!*n/256. - Vaclav Kotesovec, Apr 25 2017