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A285239
Number of entries in the n-th cycles of all permutations of [2n].
3
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 15 2017
STATUS
approved