This paper presents a combinatorial study of sums of integer powers of the cotangent. Our main tool is the realization of the cotangent values as eigenvalues of a simple self-adjoint matrix with complex integer entries. We use the trace method to draw conclusions about integer values of the sums and series expansions of the generating function to provide explicit evaluations; it is remarkable that throughout the calculations the combinatorics are governed by the higher tangent and arctangent numbers exclusively. Finally, we indicate a new approximation of the values of the Riemann zeta function at even integer arguments.

中文翻译：

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余切和的跟踪方法

本文提出了余切数的整数幂之和的组合研究。我们的主要工具是将余切值实现为具有复杂整数项的简单自伴矩阵的特征值。我们使用跟踪方法得出有关总和的整数值和生成函数的级数展开的结论，以提供明确的评估。值得注意的是，在整个计算中，组合运算仅受较高的切线和反正切数支配。最后，我们指出偶数整数参数处Riemann zeta函数值的新近似值。