A285239 Number of entries in the n-th cycles of all permutations of [2n].

3, 27, 463, 12217, 441383, 20338679, 1141073295, 75473055841, 5748862140283, 495446888127507, 47648289796265871, 5057570671179281161, 587173799850231036207, 74005641366738437835967, 10062023872139208015273375, 1467822867614662009540883265

Offset: 1

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.

All terms are odd.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..326
• Wikipedia, Permutation

Crossrefs

Cf. 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))((j-1)!
      *b(n-j, max(0, i-1)))*binomial(n-1, j-1), j=1..n)))
    end:
a:= n-> coeff(b(2*n, n), x, 1):
seq(a(n), n=1..20);

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]][(j - 1)! b[n - j, Max[0, i - 1]]] Binomial[ n - 1, j - 1], {j, 1, n}]]];
a[n_] := Coefficient[b[2n, n], x, 1];
Array[a, 20] (* Jean-François Alcover, Jun 01 2018, from Maple *)

Formula

a(n) = A185105(2n,n).

a(n) ~ 2^(3*n-1) * c^(2*n) * n^(n - 1/2) / (sqrt(Pi*(c-1)) * (2*c-1)^n * exp(n)), where c = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... - Vaclav Kotesovec, Apr 15 2017, updated Mar 10 2020