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

1,1

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

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

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 *)

Crossrefs

Cf. A185105.

Sequence in context: A193541 A193544 A286306 * A111844 A277352 A118714

Adjacent sequences: A285236 A285237 A285238 * A285240 A285241 A285242

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2017

Status

approved