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
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Apr 19 2009
STATUS
approved