A285235 Number of entries in the seventh cycles of all permutations of [n].

1, 30, 622, 11378, 199809, 3499572, 62333543, 1141073295, 21593291506, 423749322362, 8637159909596, 182967605341204, 4028364756058464, 92147187469290768, 2188667860854515856, 53939340317601471888, 1378181549321980128288, 36476226109960185948768

Offset

7,2

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 = 7..450

Wikipedia, Permutation

Formula

a(n) = A185105(n,7).

Recurrence: (n-7)*(n-4)*a(n) = (n-3)*(6*n^2 - 67*n + 176)*a(n-1) - 5*(n-4)*(3*n^3 - 43*n^2 + 195*n - 283)*a(n-2) + 10*(2*n^5 - 47*n^4 + 436*n^3 - 1999*n^2 + 4532*n - 4062)*a(n-3) - (15*n^6 - 445*n^5 + 5465*n^4 - 35555*n^3 + 129161*n^2 - 248111*n + 196528)*a(n-4) + (6*n^7 - 221*n^6 + 3473*n^5 - 30165*n^4 + 156251*n^3 - 482105*n^2 + 819087*n - 589808)*a(n-5) - (n-6)^7*(n-3)*a(n-6), for n>7. - Vaclav Kotesovec, Apr 25 2017

a(n) ~ n!*n/128. - 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, 7), x, 1):
seq(a(n), n=7..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, 7], x, 1];
Table[a[n], {n, 7, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

Crossrefs

Column k=7 of A185105.

Sequence in context: A279870 A124099 A028258 * A075911 A001719 A004359

Adjacent sequences: A285232 A285233 A285234 * A285236 A285237 A285238

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2017

Status

approved