Inspired by [this MO question about integrating binomial coefficients](https://mathoverflow.net/questions/254881/binomial-again-and-again/) and the answers, I was wondering whether integrating powers of binomial coefficients also relates to the respective sums. And indeed I have strong numerical evidence that 
$$\int\_{-\infty}^{\infty} \binom{n}{x}^2 dx =\sum\_{k=0}^n\binom{n}{k}^2$$. (The latter expression is of course $\binom{2n}{n}$.) 


So I'm wondering for which $l$ the following identity holds: 
$$\int\_{-\infty}^{\infty} \binom{n}{x}^l dx =\sum\_{k=0}^n\binom{n}{k}^l$$. 


(And furthermore one could conjecture that there are similar examples, where sum over binomials is identical to integral of the same expression over real numbers.) 


EDIT: 
Regarding the last sentence here an example: I conjecture (and have numerical evidence) that 
$$\int\_{-\infty}^{\infty} x\binom{n}{x} dx =\sum\_{k=0}^n k\binom{n}{k},$$ the latter expression being of course $2^{n-1}n$.


EDIT2:
And for Vandermonde's identity it seems also (by numerical evidence) to work analogously:
$$ \int\_{-\infty}^{\infty} \binom{m}{x} \binom{n}{r-x} dx = \sum\_{k=0}^r \binom{m}{k} \binom{n}{r-k},$$ the latter expression being of course $\binom{m+n}{r}$. 


I dare to conjecture that one can still find more examples. 



Integral of power of binomials equal to sum of power of binomials?
The generalization looks like this
$$
\int\_{-\infty}^{\infty} \binom{n}{\alpha x}^l dx =\sum\_{k=-\infty}^\infty\binom{n}{\alpha k}^l,\quad 0<\alpha\le 2/l,~l\in\mathbb{N}\tag{1}
$$
where $n$ need not be an integer. The general theorem is given for example in the paper [Surprising sinc sums and integrals](https://www.jstor.org/stable/27642636).
Below I give the general outline of the proof which is based on the well known fact that the following function is band limited (its Fourier transform has limited spectrum)
$$g(x)=\binom{n}{x}=\frac{1}{2\pi}\int\_{-\pi}^{\pi} (1+e^{i \omega})^n e^{- ix\omega} d\omega$$
One can see that Fourier transform is limited to frequencies $|\omega|<\pi$. Whenever spectrum of a function $f(x)$ is limited to frequencies $|\omega|<2\pi$ one expects that 
$$
\int\_{-\infty}^{\infty} f(x) dx =\sum\_{k=-\infty}^\infty f(k).
$$


Now the Fourier transform of $g(\alpha x)^l$ has a spectrum limited in the band $|\omega|<\pi\alpha l$. This is easy to see calculating Fourier transform
$$
\int\_{-\infty}^{\infty}g(\alpha x)^le^{-ikx}dx
$$
with the help of $\int\_{-\infty}^{\infty}e^{-ikx}dx=2\pi\delta(k)$, where $\delta$ is delta function. The general theorem from the paper cited above now states that
$$
\int\_{-\infty}^{\infty} g(\alpha x)^l dx =\sum\_{k=-\infty}^\infty g(\alpha k)^l,\quad 0<\pi\alpha l\le 2\pi,~l\in\mathbb{N},
$$
which is equivalent to (1).


All this analysis also explains that when $\alpha=1$ the proposed identity holds for $l=1,2$ but not for larger $l\in\mathbb{N}$.


