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

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 = 7..450
• Wikipedia, Permutation

Crossrefs

Column k=7 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, 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 *)

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