I considered a related question in Theorem 6 of this paper. Although it's formulated differently (in terms of random circulant matrices), my result estimates the fraction of $\pm 1$ polynomials of degree $d$ which are divisible by some cyclotomic polynomial of order dividing $d+1$. If $d$ is odd, this fraction is of the order $d^{-1/2}$. If $d$ is even, I derived two different upper bounds which imply that the fraction is smaller than in the even case; in particular, if $d+1$ is prime then $\pm \sum_{k=0}^d x^k$ are (pretty obviously) the only such polynomials.

This is clearly all closely related to the observations in Greg's answer.

Edit: Corrected a few effects of typing without thinking/reading.

I considered a related question in Theorem 6 of this paper. Although it's formulated differently (in terms of random circulant matrices), my result estimates the fraction of $\pm 1$ polynomials of degree $d$ which are divisible by some cyclotomic polynomial of order dividing $d+1$. If $d$ is odd, this fraction is of the order $d^{-1/2}$. If $d$ is even, I derived two different upper bounds which imply that the fraction is smaller than in the even case; in particular, if $d+1$ is prime then $\pm \sum_{k=0}^d x^k$ are (pretty obviously) the only such polynomials.

This is clearly all closely related to the observations in Greg's answer.

Edit: Corrected a few effects of typing without thinking/reading.

I considered a related question in Theorem 6 of this paper. Although it's formulated differently (in terms of random circulant matrices), my result estimates the fraction of $\pm 1$ polynomials of degree $d$ which are divisible by the $d$th cyclotomic polynomial. If $d$ is even, this fraction is of the order $d^{-1/2}$. If $d\ge 3$ is odd, I derived two different upper bounds which imply that the fraction is smaller than in the even case; in particular, if $d$ is prime then $\pm \sum_{k=0}^d x^k$ are (pretty obviously) the only such polynomials.

This is clearly all closely related to the observations in Greg's answer.

I considered a related question in Theorem 6 of this paper. Although it's formulated differently (in terms of random circulant matrices), my result estimates the fraction of $\pm 1$ polynomials of degree $d$ which are divisible by some cyclotomic polynomial of order dividing $d+1$. If $d$ is odd, this fraction is of the order $d^{-1/2}$. If $d$ is even, I derived two different upper bounds which imply that the fraction is smaller than in the even case; in particular, if $d+1$ is prime then $\pm \sum_{k=0}^d x^k$ are (pretty obviously) the only such polynomials.

This is clearly all closely related to the observations in Greg's answer.

Edit: Corrected a few effects of typing without thinking/reading.