Suggestion for a sequence: weights on a circle

[Brendan McKay]

A recent puzzle in New Scientist suggests the following.

Suppose you have n objects, with weights 1,2,3,...,n.
(That is, exactly one object with each of those weights.)

How many ways are there to place these objects evenly spaced
around the circumference of a disk so that the disk will
exactly balance on the centre point?

In algebraic terms, how many permutations p1,p2,...,pn
are such that the polynomial p1 + p2*x + ... + pn*x^(n-1)
has exp(2 pi i/n) as a zero?  

An example for n=10 is [2, 9, 1, 8, 5, 7, 4, 6, 3, 10]. I don't even
know which n have a solution. Certainly not n=1,2,3,4. For n=10
there are 480 solutions, or 48 equivalence classes if "1" is placed
in a fixed position, or 24 equivalence classes if the mirror
image is regarded as the same.

Brendan.