For the circulant matrix $C$ of  order $n=4$ with first row $[-1,1,1,1]$ say
$$
C = Circ(-1,1,1,1)
$$
we have the equality of vectors
$$
[R(1),R(\omega),R(\omega^2),R(\omega^3)] = c [-1,1,1,1],
$$
where 
$$
\omega=exp(2\pi i/4)=i,
$$
$$
c =-2
$$
and $R(t)$ is the the representer polynomial of $C$
namely,
$$
R(t)=-1+t+t^2+t^3.
$$

Question:  There are other such matrices $C$ when $n =4k >4.$

More precisely:  Let $k >1$ be an integer, and  let $n=4k.$ We want a matrix $C$ such that

(a) 
$$
C = Circ(h_1, \ldots,h_n),
$$
be a non-singular circulant matrix of order $n$ with $h_i \in \lbrace -1,1 \rbrace$ for all $i=1, \ldots,n.$

and

(b)
For some nonzero integer $c \neq 0$ one has the equality of vectors
$$
[R(1),R(\omega), \ldots, R(\omega^{n-1}] = c [h_1, \ldots,h_n].
$$
where 
$$
\omega=exp(2\pi i/n),
$$
and $R(t)$ is the the representer polynomial of $C$
namely,
$$
R(t)= h_1+h_2t + \cdots + h_{n}t^{n-1}.
$$