A285237 Number of entries in the ninth cycles of all permutations of [n].

1, 47, 1434, 36792, 872511, 20014299, 455265257, 10420963144, 242208466145, 5748862140283, 139849088103596, 3494752531722564, 89838192687840304, 2377612074981717632, 64807344109730799968, 1819505580964336136560, 52611858820598185363536

Offset

9,2

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 = 9..450

Wikipedia, Permutation

Formula

a(n) = A185105(n,9).

Recurrence: (n-9)*(n-6)*a(n) = 2*(4*n^3 - 77*n^2 + 471*n - 916)*a(n-1) - 14*(2*n^4 - 49*n^3 + 439*n^2 - 1714*n + 2474)*a(n-2) + 14*(n-5)*(4*n^4 - 103*n^3 + 981*n^2 - 4117*n + 6454)*a(n-3) - 7*(n-6)*(10*n^5 - 320*n^4 + 4070*n^3 - 25770*n^2 + 81333*n - 102427)*a(n-4) + 14*(4*n^7 - 185*n^6 + 3661*n^5 - 40195*n^4 + 264477*n^3 - 1042986*n^2 + 2282488*n - 2138058)*a(n-5) - (28*n^8 - 1554*n^7 + 37702*n^6 - 522242*n^5 + 4517128*n^4 - 24979724*n^3 + 86233855*n^2 - 169871843*n + 146155098)*a(n-6) + (8*n^9 - 526*n^8 + 15356*n^7 - 261226*n^6 + 2853242*n^5 - 20747608*n^4 + 100420076*n^3 - 311890495*n^2 + 563892963*n - 452026202)*a(n-7) - (n-8)^9*(n-5)*a(n-8), for n>9. - Vaclav Kotesovec, Apr 25 2017

a(n) ~ n!*n/512. - 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, 9), x, 1):
seq(a(n), n=9..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, 9], x, 1];
Table[a[n], {n, 9, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

Crossrefs

Column k=9 of A185105.

Sequence in context: A047911 A009069 A348805 * A157359 A153214 A142845

Adjacent sequences: A285234 A285235 A285236 * A285238 A285239 A285240

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2017

Status

approved