Working out the following definition of the [Cyclotomic Polynomial](https://en.wikipedia.org/wiki/Cyclotomic_polynomial)
$$ {\displaystyle \Phi \_{n}(x)=\prod \_{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right),}
$$
you'll get
$$
{\displaystyle x^{\varphi(n)}-x^{\varphi(n)-1}\sum \_{\stackrel {1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i{\frac {k}{n}}}} + \dots + 1
$$

You'll spot right away that the next to leading coefficient gives the [Möbius function](https://en.wikipedia.org/wiki/M%C3%B6bius_function):
$$ {\displaystyle \mu (n)=\sum \_{\stackrel {1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i{\frac {k}{n}}}}
$$

For the final $1$, you'll get:
$$
\prod \_{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(-e^{2i\pi {\frac {k}{n}}}\right)=\left(e^{\frac{2i\pi}{n} \sum\_{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}k}\right )\prod \_{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}(-1)=\left(e^{\frac{2i\pi}{n}\frac{n}{2}\varphi(n) }\right )(-1)^{\varphi(n)}=+1,
$$
which is expected since the coefficients are symmetric and $x^{\varphi(n)}$ also has $+1$.

> 
> Do the other terms also have number theoretical interpretation?
> 
> 
> 

UPDATE:

A first partial result found at OEIS: [Coefficient of $x^2$ in the $n$-th cyclotomic polynomial.](https://oeis.org/A086823)

*If $n$ is odd, $a(n) = 1/2 (\mu(n)^2-\mu(n))$,*
*if $n$ is even, $a(n) = 1/2 (\mu(n)^2-\mu(n)) - \mu(n/2)$, where $\mu$ is Möbius function...*