The $n^{th}$ cyclotomic polynomial ${\Phi }_n\left(x\right)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers $n$. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time!

In this paper we present a survey of these formulas which until now were scattered in the literature, and introduce a unified approach to derive some of them, leading also to shorter proofs as a by-product. In particular, we show that some of the formulas have a more elegant reinterpretation in terms of Bell polynomials. This approach amounts to computing the logarithmic derivatives of ${\Phi }_n$ at certain points. Furthermore, we show that the logarithmic derivatives at $\pm 1$ of any Kronecker polynomial (a monic product of cyclotomic polynomials and a monomial) satisfy a family of linear equations whose coefficients are Stirling numbers of the second kind. We apply these equations to show that certain polynomials are not Kronecker. In particular, we infer that for every $k\ge 4$ there exists a symmetric numerical semigroup with embedding dimension $k$ and Frobenius number $2k+1$ that is not cyclotomic, thus establishing a conjecture of Alexandru Ciolan, Pedro García-Sánchez and the second author. In an appendix Pedro García-Sánchez shows that for every $k\ge 4$ there exists a symmetric non-cyclotomic numerical semigroup having Frobenius number $2k+1.$

这 $n^{吨H}$ 分圆多项式 ${\Phi }_n\left(X\right)$ 是一个的最小多项式 $n^{吨H}$统一的原始根。它的系数是深入研究的主题，其中一些公式是已知的。这里我们对适用于所有自然数的公式感兴趣$n$. 在这些中，出现了许多著名的数论对象，例如伯努利数、斯特林数和拉马努金和，有时甚至同时出现！

在本文中，我们对迄今为止分散在文献中的这些公式进行了调查，并介绍了一种统一的方法来推导其中的一些公式，同时还导致更短的证明作为副产品。特别是，我们展示了一些公式在贝尔多项式方面有更优雅的重新解释。这种方法相当于计算的对数导数${\Phi }_n$在某些点。此外，我们证明了对数导数在$\pm 1$任何 Kronecker 多项式（分圆多项式和单项式的单数乘积）的 满足一系列线性方程，其系数是第二类斯特林数。我们应用这些方程来表明某些多项式不是 Kronecker。特别地，我们推断对于每个$克\ge 4$ 存在一个具有嵌入维数的对称数值半群 $克$ 和弗罗贝尼乌斯数 $2克+1$这不是循环的，因此建立了 Alexandru Ciolan、Pedro García-Sánchez 和第二作者的猜想。在附录中 Pedro García-Sánchez 表明，对于每个$克\ge 4$ 存在具有 Frobenius 数的对称非分圆数值半群 $2克+1.$