A159662 Any number of necklaces made from n distinct colored beads then linearly arranged in a display case.

1, 1, 3, 13, 77, 572, 5114, 53406, 637818, 8572434, 128041458, 2103949314, 37716766350, 732505270152, 15320768312784, 343335554738328, 8207083694470392, 208444177385240472, 5605513502234263272

Offset

0,3

Comments

a(n) is the number of ways to seat n people at circular tables then linearly order the tables. Two seating arrangements are considered identical if each person has the same two neighbors in both.

Links

G. C. Greubel, Table of n, a(n) for n = 0..420

Formula

E.g.f.: 1/(1 -x/2 -x^2/4 + log(1-x)/2).

a(n) ~ n! * 2*(r-1)/((r^2-2)*r^(n+1)), where r = 0.669337307032878... is the root of the equation 2*log(1-r) = r^2 + 2*r - 4. - Vaclav Kotesovec, Sep 25 2013

Example

a(3)=13 because: There are 3! ways to arrange the three necklaces consisting of a single bead. There are 2! ways to arrange each of the 3 collections of necklaces of length two and one. There is 1 way to display the unique necklace having three beads. 3!+2!*3+1=13.

Mathematica

CoefficientList[Series[1/(1 - x/2 - x^2/4 + Log[1-x]/2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}]

Prog

(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( 4/(4-2*x-x^2 +2*Log(1-x)) ))); // G. C. Greubel, Sep 27 2022
(SageMath)
def A159662_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( 4/(4-2*x-x^2 +2*log(1-x)) ).egf_to_ogf().list()
A159662_list(40) # G. C. Greubel, Sep 27 2022

Crossrefs

Cf. A001710.

Sequence in context: A336421 A189239 A074530 * A032035 A351421 A273953

Adjacent sequences: A159659 A159660 A159661 * A159663 A159664 A159665

Keyword

nonn

Author

Geoffrey Critzer, Apr 19 2009

Status

approved