Evaluating $\int_0^u\int_0^{2u-x_1}\cdots \int_0^{ku-x_1-\cdots -x_{n-1}}dx_ndx_{n-1}\cdots dx_2dx_1$ I have reduced a problem to evaluating the integral $\int_0^u\int_0^{2u-x_1}\cdots \int_0^{ku-x_1-x_2-...-x_{n-1}}dx_ndx_{n-1}\cdots dx_2dx_1$.
I tried computing this for small $n=1,2,3$ but I can't seem to find any formula that I could then prove inductively. The parameter $u$ here is a real number.

Through that integral we are computing the $k$-dimensional Lebesgue measure  of the set given by $x_i\geq 0$ and $x_1+x_2+\ldots+x_n\leq nu$ for every $n\in[1,k]$. That is obviously $u^k$ times the $k$-dimensional Lebesgue measure of the set given by $x_i\geq 0$ and $x_1+x_2+\ldots+x_n\leq n$ for every $n\in[1,k]$. Let $S_n = x_1+x_2+\ldots+x_n$. We must have $0\leq S_n\leq n$ and $S_{n+1}\geq S_n$. So we may choose at random (with respect to a uniform distribution) $k$ points from the interval $[0,k]$ and, assuming such points are $P_1\leq P_2\leq\ldots\leq P_k$, compute the probability $\mathbb{P}_k$ that such points fulfill $P_n\leq n$ for every $n\in[1,k]$. $\mathbb{P}_k$ can be computed by induction, by computing first what is the probability that the maximum of $n$ independent, uniformly distributed over $[0,1]$, random variables lies in $\left[1-\frac{1}{n},1\right]$. That probability is $1-\left(1-\frac{1}{n}\right)^n$.
At last we have:

$$ \int_{0}^{u}\int_{0}^{2u-x_1}\cdots\int_{0}^{nu-x_1-\ldots-x_{n-1}}dx_n\,dx_{n-1}\ldots dx_1 = u^n \frac{(n+1)^n}{(n+1)!}.$$

That numbers arise in the Taylor series of the Lambert W function.
