A159662 - OEIS

%I #16 Sep 27 2022 07:16:11

%S 1,1,3,13,77,572,5114,53406,637818,8572434,128041458,2103949314,

%T 37716766350,732505270152,15320768312784,343335554738328,

%U 8207083694470392,208444177385240472,5605513502234263272

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

%C 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.

%H G. C. Greubel, <a href="/A159662/b159662.txt">Table of n, a(n) for n = 0..420</a>

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

%F 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

%e 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.

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

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o Coefficients(R!(Laplace( 4/(4-2*x-x^2 +2*Log(1-x)) ))); // _G. C. Greubel_, Sep 27 2022

%o (SageMath)

%o def A159662_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( 4/(4-2*x-x^2 +2*log(1-x)) ).egf_to_ogf().list()

%o A159662_list(40) # _G. C. Greubel_, Sep 27 2022

%Y Cf. A001710.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Apr 19 2009