Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$ I found the following formula 
$$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$
and it is cited that Euler proved the formula above , but how ?
Do there exist other proofs ?
Can we have a general formula for the alternating form 
$$\sum_{n=1}^\infty (-1)^{n+1}\frac{H_n}{n^q}$$
 A: $$
\begin{align}
&\sum_{j=0}^k\zeta(k+2-j)\zeta(j+2)\\
&=\sum_{m=1}^\infty\sum_{n=1}^\infty\sum_{j=0}^k\frac1{m^{k+2-j}n^{j+2}}\tag{1}\\
&=(k+1)\zeta(k+4)
+\sum_{\substack{m,n=1\\m\ne n}}^\infty\frac1{m^2n^2}
\frac{\frac1{m^{k+1}}-\frac1{n^{k+1}}}{\frac1m-\frac1n}\tag{2}\\
&=(k+1)\zeta(k+4)
+\sum_{\substack{m,n=1\\m\ne n}}^\infty\frac1{nm^{k+2}(n-m)}-\frac1{mn^{k+2}(n-m)}\tag{3}\\
&=(k+1)\zeta(k+4)
+2\sum_{m=1}^\infty\sum_{n=m+1}^\infty\frac1{nm^{k+2}(n-m)}-\frac1{mn^{k+2}(n-m)}\tag{4}\\
&=(k+1)\zeta(k+4)
+2\sum_{m=1}^\infty\sum_{n=1}^\infty\frac1{(n+m)m^{k+2}n}-\frac1{m(n+m)^{k+2}n}\tag{5}\\
&=(k+1)\zeta(k+4)\\
&+2\sum_{m=1}^\infty\sum_{n=1}^\infty\frac1{m^{k+3}n}-\frac1{(m+n)m^{k+3}}\\
&-2\sum_{m=1}^\infty\sum_{n=1}^\infty\frac1{m(n+m)^{k+3}}+\frac1{n(n+m)^{k+3}}\tag{6}\\
&=(k+1)\zeta(k+4)
+2\sum_{m=1}^\infty\frac{H_m}{m^{k+3}}
-4\sum_{n=1}^\infty\sum_{m=1}^\infty\frac1{n(n+m)^{k+3}}\tag{7}\\
&=(k+1)\zeta(k+4)
+2\sum_{m=1}^\infty\frac{H_m}{m^{k+3}}
-4\sum_{n=1}^\infty\sum_{m=n+1}^\infty\frac1{nm^{k+3}}\tag{8}\\
&=(k+1)\zeta(k+4)
+2\sum_{m=1}^\infty\frac{H_m}{m^{k+3}}
-4\sum_{n=1}^\infty\sum_{m=n}^\infty\frac1{nm^{k+3}}+4\zeta(k+4)\tag{9}\\
&=(k+5)\zeta(k+4)
+2\sum_{m=1}^\infty\frac{H_m}{m^{k+3}}
-4\sum_{m=1}^\infty\sum_{n=1}^m\frac1{nm^{k+3}}\tag{10}\\
&=(k+5)\zeta(k+4)
+2\sum_{m=1}^\infty\frac{H_m}{m^{k+3}}
-4\sum_{m=1}^\infty\frac{H_m}{m^{k+3}}\tag{11}\\
&=(k+5)\zeta(k+4)
-2\sum_{m=1}^\infty\frac{H_m}{m^{k+3}}\tag{12}
\end{align}
$$
Letting $q=k+3$ and reindexing $j\mapsto j-1$ yields
$$
\sum_{j=1}^{q-2}\zeta(q-j)\zeta(j+1)
=(q+2)\zeta(q+1)-2\sum_{m=1}^\infty\frac{H_m}{m^q}\tag{13}
$$
and finally
$$
\sum_{m=1}^\infty\frac{H_m}{m^q}
=\frac{q+2}{2}\zeta(q+1)-\frac12\sum_{j=1}^{q-2}\zeta(q-j)\zeta(j+1)\tag{14}
$$

Explanation
$\hphantom{0}(1)$ expand $\zeta$
$\hphantom{0}(2)$ pull out the terms for $m=n$ and use the formula for finite geometric sums on the rest
$\hphantom{0}(3)$ simplify terms
$\hphantom{0}(4)$ utilize the symmetry of $\frac1{nm^{k+2}(n-m)}+\frac1{mn^{k+2}(m-n)}$
$\hphantom{0}(5)$ $n\mapsto n+m$ and change the order of summation
$\hphantom{0}(6)$ $\frac1{mn}=\frac1{m(m+n)}+\frac1{n(m+n)}$
$\hphantom{0}(7)$ $H_m=\sum_{n=1}^\infty\frac1n-\frac1{n+m}$ and use the symmetry of $\frac1{m(n+m)^{k+3}}+\frac1{n(n+m)^{k+3}}$
$\hphantom{0}(8)$ $m\mapsto m-n$
$\hphantom{0}(9)$ subtract and add the terms for $m=n$
$(10)$ combine $\zeta(k+4)$ and change the order of summation
$(11)$ $H_m=\sum_{n=1}^m\frac1n$
$(12)$ combine sums  
A: Although this problem is from April 2013 I would like to take it up and try to complete the answer turning to the question
"Can we have a general formula for the alternating form?"
$$S_a(q) = \sum_{n=1}^\infty (-1)^{n+1}\frac{H_n}{n^q}$$
By inspecting the first various expressions I have made the following guess for the alternating series for even $q = 2, 4, ...$   
$$S_a(q=2,4,...) = c(q)\frac{ \zeta (q+1)}{2^{q+1}}-\sum _{k=1}^{\frac{q}{2}-1} \left(1-\frac{1}{2^{q-2 k-1}}\right) \zeta (2 k+1) \zeta (q-2 k)\tag{1}$$
Here $c(q)$ are coefficients. The first 10 entries are
$$c(2,4,..,20) = \{5,59,377,2039,10229,49139,229361,1048559,4718573,20971499\}\tag{1a}$$
This sequence is not contained in https://oeis.org and I could not find a formula up to now. 
For odd $q$ Mathematica returns a seemingly simple pattern
$$S_a(q=1)= \frac{\pi ^2}{12}-\frac{\log ^2(2)}{2}\tag{2a}$$
$$S_a(q=3,5,...)= \gamma  \left(1-\frac{1}{2^{q-1}}\right) \zeta (q)-\;{_aF}_b^{reg}(q)\tag{2b}$$
where $\gamma$ is the Euler gamma, and ${_ aF}_b^{reg}(q)$ is the partial derivative of the regularized hypergeometric function
 with the parameter sets $a$ and $b$ with repect to the last parameter in $b$ taken at the argument -1. 
I still need to understand this function better before posting it here. Most probably it hides a pattern similar to that of (1).
EDIT
After having completed the entry up to this point I found that the case of odd $q$ has already been treated extensively in Calculating alternating Euler sums of odd powers in March 2017.
Using these results we can easily identify the coefficients (1a) as
$$c(q) = q \left(2^q-1\right)-1$$
A: Answering the first part of the question for $q$ odd we recall from the following MSE post the identity:
$$ H_n = - \frac{1}{2\pi i} \int_{-1/2-i\infty}^{-1/2+i\infty} \zeta(1-s) 
\frac{\pi}{\sin(\pi s)}\frac{1}{n^s} ds.$$
The proof at the above cited post is sound and I will merely refer to it here since otherwise we would just include it verbatim.
This gives the formula for your sum:
$$\sum_{n\ge 1} \frac{H_n}{n^q} =
-  \frac{1}{2\pi i} \int_{-1/2-i\infty}^{-1/2+i\infty} \zeta(1-s) 
\frac{\pi}{\sin(\pi s)} \zeta(q+s) ds.$$
Now shift this integral to the left to the line $\Re(s) = -1/2-(q-1),$ getting
$$\sum_{n\ge 1} \frac{H_n}{n^q} = \rho_1
- \sum_{k=1}^{q-2} \zeta(1+k) (-1)^k \zeta(q-k)
- \frac{1}{2\pi i} \int_{-1/2-(q-1)-i\infty}^{-1/2-(q-1)+i\infty} \zeta(1-s) 
\frac{\pi}{\sin(\pi s)} \zeta(q+s) ds$$
where $$\rho_1 = 
\operatorname{Res}\left( -\zeta(1-s) 
\frac{\pi}{\sin(\pi s)} \zeta(q+s); s=-(q-1)\right).$$
Make the substitution $t=s+(q-1)$ in the integral to get (not including the minus sign in front)
$$  \frac{1}{2\pi i}\int_{-1/2-i\infty}^{-1/2+i\infty} \zeta(1-(t-(q-1))) 
\frac{\pi}{\sin(\pi (t-(q-1))} \zeta(q+t-(q-1)) dt.$$
For $q$ odd this simplifies to
$$ \frac{1}{2\pi i}\int_{-1/2-i\infty}^{-1/2+i\infty} \zeta(q-t) 
\frac{\pi}{\sin(\pi t)} \zeta(t+1) dt.$$
Now make another substitution, namely $v=-t$, to get
$$ \frac{1}{2\pi i}\int_{1/2+i\infty}^{1/2-i\infty} \zeta(q+v) 
\frac{\pi}{\sin(\pi v)} \zeta(1-v) dv
=-\frac{1}{2\pi i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(q+v) 
\frac{\pi}{\sin(\pi v)} \zeta(1-v) dv$$
where the minus on the sine term cancels the one on the differential.
Finally shift this integral to the line $\Re(v) = -1/2$ to obtain
$$\rho_2 - \frac{1}{2\pi i}\int_{-1/2-i\infty}^{-1/2+i\infty} \zeta(q+v) 
\frac{\pi}{\sin(\pi v)} \zeta(1-v) dv
= \rho_2 + \sum_{n\ge 1} \frac{H_n}{n^q}$$
where $$\rho_2 = 
\operatorname{Res}\left(- \zeta(1-v) 
\frac{\pi}{\sin(\pi v)} \zeta(q+v); v=0\right).$$
We have shown that
$$\sum_{n\ge 1} \frac{H_n}{n^q} = 
\rho_1 - \sum_{k=1}^{q-2} \zeta(1+k) (-1)^k \zeta(q-k) - 
\left(\rho_2 + \sum_{n\ge 1} \frac{H_n}{n^q}\right).$$
This gives
$$ \sum_{n\ge 1} \frac{H_n}{n^q} = \frac{1}{2} (\rho_1-\rho_2)
- \frac{1}{2} \sum_{k=1}^{q-2} \zeta(1+k) (-1)^k \zeta(q-k).$$
To conclude introduce
$$ W(s) = -\zeta(1-s) 
\frac{\pi}{\sin(\pi s)} \zeta(q+s).$$
This implies that
$$ W(-s-(q-1)) = -\zeta(s+q) \frac{\pi}{\sin(\pi (-s-(q-1)))} 
\zeta(1-s) = - W(s)$$
because $q$ is odd.
Now $$\rho_2 = \frac{1}{2\pi i} \int_{|s|=1/2} W(s) ds.$$ 
Put $s = -t -(q-1)$ and note that this does not change the counterclockwise orientation of the circle induced by the first integral to get
$$ -\frac{1}{2\pi i} \int_{|-t-(q-1)|=1/2} W(-t-(q-1)) dt
= \frac{1}{2\pi i} \int_{|-t-(q-1)|=1/2} W(t) dt = \rho_1$$
because $|-t-(q-1)|=|(-1)(t+(q-1))|=|t-(-(q-1))|.$
The conclusion is that
$$ \sum_{n\ge 1} \frac{H_n}{n^q} = 
-\frac{1}{2} \sum_{k=1}^{q-2} \zeta(1+k) (-1)^k \zeta(q-k)$$
for $q$ odd.
Addendum. Sun Apr 27 23:57:35 CEST 2014 I don't quite see why I didn't simply evaluate the residues $\rho_1$ and $\rho_2$ as these are both easy. This does not affect the correctness of the argument.
Addendum. Sun Nov  9 23:33:24 CET 2014 In fact the equality of the two residues follows by inspection. In retrospect it appears I wanted to avoid working with the two double poles and keep everything within the limits of pen and paper.
A: We have:
\begin{eqnarray}
\sum\limits_{n=1}^\infty \frac{H_n}{n^q} &=& \sum\limits_{n=1}^\infty \frac{H_n}{(n+1)^q} + \zeta(q+1) \\
&=& 1/2 \left(q \zeta(q+1) - \sum\limits_{j=1}^{q-2} \zeta(j+1) \zeta(q-j) \right)+ \zeta(q+1)
\end{eqnarray}
where in the last line we used the result given in the answer to question Closed form expressions for harmonic sums .
A: Partial solution:
I am going to prove 

$$\sum_{k=1}^\infty\frac{H_k}{k^n}=\frac12\sum_{i=1}^{n-2}(-1)^{i-1}\zeta(n-i)\zeta(i+1),\quad n=3,5,7, ...$$


We have 
$$\int_0^1x^{k-1}\operatorname{Li}_n(x)\ dx\overset{IBP}{=}(-1)^{n-1}\frac{H_k}{k^n}-\sum_{i=1}^{n-1}(-1)^i\frac{\zeta(n-i+1)}{k^i}$$
Divide both sides by $k$ then consider the summation from $k=1$ to $\infty$ we have
$$\int_0^1\frac{\operatorname{Li}_n(x)}{x}\sum_{k=1}^\infty\frac{x^k}{k}\ dx=(-1)^{n-1}\sum_{k=1}^\infty\frac{H_k}{k^{n+1}}-\sum_{i=1}^{n-1}(-1)^i\zeta(n-i+1)\sum_{k=1}^\infty\frac1{k^{i+1}}$$
$$\small{-\int_0^1\frac{\operatorname{Li}_n(x)\ln(1-x)}{x}\ dx=(-1)^{n-1}\sum_{k=1}^\infty\frac{H_k}{k^{n+1}}-\sum_{i=1}^{n-1}(-1)^i\zeta(n-i+1)\zeta(i+1)}\tag1$$
where  
$$-\int_0^1\frac{\operatorname{Li}_n(x)\ln(1-x)}{x}\ dx=-\sum_{k=1}^\infty\frac1{k^n}\int_0^1 x^{k-1}\ln(1-x)\ dx=\sum_{k=1}^\infty\frac{H_k}{k^{n+1}}\tag2$$
Plug (2) in (1) we get
$$\sum_{k=1}^\infty\frac{H_k}{k^{n+1}}[1+(-1)^n]=-\sum_{i=1}^{n-1}(-1)^i\zeta(n-i+1)\zeta(i+1)$$
Let $n-1\mapsto n$ to get
$$\sum_{k=1}^\infty\frac{H_k}{k^{n}}[1-(-1)^n]=-\sum_{i=1}^{n-2}(-1)^i\zeta(n-i)\zeta(i+1)$$
So clearly for odd $n\geq3$ we have 
$$\sum_{k=1}^\infty\frac{H_k}{k^{n}}[2]=-\sum_{i=1}^{n-2}(-1)^i\zeta(n-i)\zeta(i+1)$$
set $n=2m+1$
$$\sum_{k=1}^\infty\frac{H_k}{k^{2m+1}}=-\frac12\sum_{i=1}^{2m-1}(-1)^i\zeta(2m+1-i)\zeta(i+1),\quad m=1,2,3,...$$
A: When $q$ is odd and greater than $1$, one can show $$ \sum_{n=1}^{\infty} \frac{H_{n}}{n^{q}} = \frac{1}{2} \sum_{k=1}^{q-2} (-1)^{k-1} \zeta(k+1) \zeta(q-k)$$
by replacing $H_{n}$ with the integral representation 
$$ H_{n} = \int_{0}^{1} \frac{1-x^{n}}{1-x} \, dx \ ,$$
switching the order of integration and summation, and then repeatedly integrating by parts.
This result is also derived in Marko Riedel's answer using a different approach.

$$ \begin{align} \sum_{n=1}^{\infty} \frac{H_{n}}{n^{q}} &= \sum_{n=1}^{\infty} \frac{1}{n^{q}} \int_{0}^{1} \frac{1-x^{n}}{1-x} \, dx \\ &= \int_{0}^{1} \frac{1}{1-x} \sum_{n=1}^{\infty} \frac{1-x^{n}}{n^{q}} \, dx \\ &= \int_{0}^{1} \frac{\zeta(q)- \text{Li}_{q}(x)}{1-x} \, dx \\ &= - \Big(\zeta(q) -  \text{Li}_{q}(x) \Big) \ln(1-x) \Bigg|^{1}_{0} - \int_{0}^{1} \frac{\log(1-x) \text{Li}_{q-1}(x)}{x} \, dx \\ &= -\color{#C00000} {\int_{0}^{1} \frac{\log(1-x) \text{Li}_{q-1}(x)}{x} \, dx} \\ &= \text{Li}_{2}(x) \text{Li}_{q-1}(x) \Bigg|^{1}_{0} - \int_{0}^{1} \frac{\text{Li}_{2}(x) \text{Li}_{q-2}(x)}{x} \, dx \\ &= \zeta(2) \zeta(q-1) - \int_{0}^{1} \frac{\text{Li}_{2}(x) \text{Li}_{q-2}(x)}{x} \, dx \\ &= \zeta(2) \zeta(q-1) - \text{Li}_{3}(x) \text{Li}_{q-2}(x) \Bigg|^{1}_{0} + \int_{0}^{1} \frac{\text{Li}_{3}(x)\text{Li}_{q-3}(x) }{x} \, dx \\ &= \zeta(2) \zeta(q-1) - \zeta(3) \zeta(q-2) + \int_{0}^{1} \frac{\text{Li}_{3}(x)\text{Li}_{q-3}(x) }{x} \, dx \\&= \zeta(2) \zeta(q-1) - \zeta(3) \zeta(q-2) + \zeta(4) \zeta(q-3) - \int_{0}^{1} \frac{\text{Li}_{4}(x) \text{Li}_{4-q}(x)}{x} \, dx \\ &=\zeta(2) \zeta(q-1) - \zeta(3) \zeta(q-2) + \zeta(4) \zeta(q-3) - \ldots + \zeta(q-1) \zeta(2) - \int_{0}^{1} \frac{\text{Li}_{q-1}(x) \text{Li}_{1}(x)}{x} \, dx \\  &= \sum_{k=1}^{q-2} (-1)^{k-1} \zeta(k+1) \zeta(q-k) + \color{#C00000}{\int_{0}^{1} \frac{\log(1-x) \text{Li}_{q-1}(x)}{x} \, dx} \end{align}$$
Therefore, if $q$ is odd,
$$\sum_{n=1}^{\infty} \frac{H_{n}}{n^{q}} = - \int_{0}^{1} \frac{\log(1-x) \text{Li}_{q-1}(x)}{x} \, dx = \frac{1}{2} \sum_{k=1}^{q-2} (-1)^{k-1} \zeta(k+1) \zeta(q-k).$$
A: Note that,      
$\displaystyle \int_{0}^{1} x^{n-1} \mathrm{d}x = \dfrac{1}{n}$   
Differentiating w.r.t. to $n$, $(p-1)$ times, we get,  
$\displaystyle \dfrac{1}{n^{p}} = \dfrac{(-1)^{p-1}}{(p-1)!} \int_{0}^{1} x^{n-1} [\ln(x)]^{p-1} \mathrm{d}x$  
$\displaystyle \implies \text{S} = \sum_{n=1}^{\infty} \dfrac{H_{n}}{n^{p}} = \dfrac{(-1)^{p-1}}{(p-1)!} \int_{0}^{1} [\ln(x)]^{p-1} \sum_{n=1}^{\infty} H_{n} x^{n-1} \mathrm{d}x $    
Since $\displaystyle \sum_{n=1}^{\infty} H_{n} x^{n} = -\dfrac{\ln(1-x)}{1-x} $, we get,     
$\displaystyle \text{S} =  \dfrac{(-1)^{p}}{(p-1)!} \int_{0}^{1}\dfrac{[\ln(x)]^{p-1} \cdot \ln(1-x) }{x(1-x)} \mathrm{d}x   $  
Recall the Beta Function $\displaystyle \operatorname{B}(a,b) = \int_{0}^{1} x^{a-1} (1-x)^{b-1} \mathrm{d}x  = \dfrac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}$  
$\displaystyle \implies \text{S} = \dfrac{(-1)^{p}}{(p-1)!} \lim_{a \to 0^+} \lim_{b \to 0^+} \left(\dfrac{{\partial}^{p-1}}{\partial a^{p-1}} \left( \dfrac{\partial}{\partial b} \operatorname{B}(a,b) \right)\right)  $   
$\therefore \displaystyle \sum_{n=1}^{\infty} \dfrac{H_{n}}{n^{p}} = \left(1+\dfrac{p}{2} \right)\zeta(p+1)-\dfrac{1}{2}\sum_{k=1}^{p-2}\zeta(k+1)\zeta(p-k)$.   
This is valid for any integer $p \geq 2$.
