Consider an hypercube in $N$ dimensions with one vertex in the origin and another vertex in the point with all coordinates equal to 1, with edges alignes with axes. Is there a general analytic expression, close form, not in terms of an integral or of an intinite series, for the function $f(x)$ that measures the volume of the part of the hypercube where the sum of the $N$ coordinates is larger than $x$, as a function of $x$ and $N$? In other words
\begin{equation} f_N(x) = \int_0^1 dx_1 \int_0^1 dx_2 ... \int_0^1 dx_N \Theta\left(x_1+x_2+...+x_n-x\right) \end{equation}
My first guess would be no, but I am reading on a paper whose result, if correct, would imply a yes to this question.
The desired volume is naturally equivalent to the probability that the Irwin–Hall distribution on $N$ $U(0,1)$ random variables is larger than $x$, or equivalently smaller than $N-x$. The CDF is given in the link: $$P(X\le x)=\frac1{N!}\sum_{k=0}^{\lfloor x\rfloor}(-1)^k\binom Nk(x-k)^N$$ Replace $x$ with $N-x$ and you have your formula.