For questions about and related to so-called Euler Sums, that are sums having [tag:harmonic-numbers] and negative integer powers of the index as coefficients.

Classical Euler Sums are of the form

$$s(p,q)~=~\sum_{n=1}^{\infty}\frac{H_n^{(p)}}{n^q}$$

for integers $p,q$ where $H_n^{(p)}$ is a generalized harmonic number. The value $p+q$ is called the *weight* of the Euler Sum. Closed-forms are known for $p=1$ and $q\geq2$, for $p=q$ and $p+q\geq4$, for $p+q\geq5$ and $q\geq2$ and for some exceptional cases.

For $p=1$ and $q\geq2$ we have Euler's formula

$$\sum_{n=1}^\infty\frac{H_n}{n^q}~=~\left(1+\frac q2\right)\zeta(q+1)+\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$

For $p=q$ and $p+q\geq4$ we have

$$\sum_{n=1}^\infty\frac{H_n^{(p)}}{n^p}~=~\frac12\left(\zeta(2p)+\zeta^2(p)\right)$$

Inconsistently, generalizations as attaching extra coefficients, e.g. $2^{-n}$, or an alternating sign $(-1)^n$ are also referred to as Euler Sums.

**References**