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

8,2

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

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

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 *)

Crossrefs

Column k=8 of A185105.

Sequence in context: A028224 A028218 A028199 * A208949 A255496 A098612

Adjacent sequences: A285233 A285234 A285235 * A285237 A285238 A285239

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2017

Status

approved