TITLE: Binomial again, and again
QUESTION [37 upvotes]: Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the ceiling function. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$.
Recently, I posted a problem on MO about "sin-omials". Here is another one in the same spirit. 

QUESTION. Numerical evidence suggest that 
  $$\left\lceil \int_0^n\binom{n}x dx\right\rceil=\sum_{k=0}^n\binom{n}k.$$
  If true, how do we prove this?

REMARK. The RHS evaluates to $2^n$; I only want to stress parallelism between the integral and sum.

REPLY [5 votes]: The $2^n$ result in Noam Elkies answer generalizes to $n=s$ for $real(s)>-1$, as he notes in MSE. In this range, the generalized binomial coefficient $ \binom{s}{\alpha}$ is a Fourier band-limited fct. in $\alpha$, or, equivalently a sinc fct. interpolation of the sequence $\binom{s}{n}$. See my notes "Fractional calculus and interpolation of generalized binomial coefficients" for simple derivations and my contribution to the MO-Q Generalizing a problem to make it easier for the integral of the sinc fct. over infinte limits, also simply derived.
The Wikipedia article on the classic Gibbs phenomena explores the extrema of $Si(x)$, easily visualized as a convolution of the sinc with the rectangle function, and, therefore, bounds on the residual of the ceiling.
The sinc function interpolation is an important illustration of the Nyquist-Shannon sampling theorem and a special case of a generalized Chu-Vandermonde identity using Euler's reflection formula for the gamma/factorial function. (I remember long ago reading somewhere that Ramanujan noted this. Anyone know a ref? This would be simple check for his Master Theorem/Formula.)