How to simplify $\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$

Question

Let $$x=\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$$ where $n,k\in\mathbb{Z}^+$. How to simplify $x$?

I simplified it for $k=1,2,3$ and I got $n$, $\dfrac12n(n+1)$ and $\dfrac16n(n+1)(n+2)$. From this I assumed that for any $k$: $$x=\dfrac n{k+1}{\binom{n+k}{n}}$$ Problem is how to prove it. I tried using mathematical induction, but it seems that this is not an easy way.