TITLE: Closed form for $\int_0^1 \int_0^1 {(1-xy)}^n \, dx \, dy$
QUESTION [5 upvotes]: I was trying to calculate 
$$\sum_{k=0}^n (-1)^k \binom{n}{k} \frac{1}{(n+1-k)^2}$$
and I found that it suffices to calculate the 
$$I_n=\int_0^1 \int_0^1 {(1-xy)}^n \,dx\,dy.$$
Is there any way to find a closed form for this? I tried to use the ordinary and the exponential generating functions of $\{I_n\}_{n\in \mathbb{N}}$, but I couldn't proceed. I also tried the change of variables 
$$x=\frac{u+v}{2},\ \ \ y=\frac{u-v}{2},$$
but it got me nowhere. Any hints or ideas?
Thanks in advance for your help.

REPLY [7 votes]: $$\begin{eqnarray*}\iint_{(0,1)^2}(1-xy)^n\,dx\,dy &\stackrel{\text{Symmetry}}{=}&2\int_{0}^{1}\int_{0}^{x}(1-xy)^n\,dy\,dx\\&\stackrel{y\mapsto x z}{=}&2 \int_{0}^{1}\int_{0}^{1}x(1-x^2 z)^n\,dz\,dx\\&\stackrel{\text{Fubini}}{=}&\frac{1}{n+1}\int_{0}^{1}\frac{1-(1-z)^{n+1}}{z}\,dz\\&\stackrel{z\mapsto 1-t}{=}&\frac{1}{n+1}\int_{0}^{1}\frac{1-t^{n+1}}{1-t}\,dt = \color{red}{\frac{H_{n+1}}{n+1}}.\end{eqnarray*}$$

As an alternative:
$$\begin{eqnarray*}\sum_{k=0}^{n}(-1)^k\binom{n}{k}\frac{1}{(n+1-k)^2}&=&\sum_{k=0}^{n}(-1)^k\binom{n}{k}\int_{0}^{1}x^{n-k}\left(-\log x\right)\,dx\\
&=& \int_{0}^{1}(-\log x)\sum_{k=0}^{n}(-1)^k\binom{n}{k}x^{n-k}\,dx\\&=& (-1)^{n+1}\int_{0}^{1}(1-x)^n\log(x)\,dx\end{eqnarray*}$$
where
$$ \int_{0}^{1}x^n\log(1-x)\,dx \stackrel{\text{IBP}}{=}-\frac{1}{n+1}\int_{0}^{1}\frac{1-x^{n+1}}{1-x}\,dx $$
and the conclusion is the same as before.