Let $\mathcal{M}(n,k) := \{ \, m\in\mathbb{N}^n \,\,\lvert \,\,\, \sum_{i=1}^n m_i = k \,\}$, where $\mathbb{N}$ denotes the natural numbers with the $0$. Also, let $M(n,k) := |\mathcal{M}(n,k)|$ denote the number of elements in this set.
I assume that the numbers $M(n,k)$ are well known. I'm looking for a general formula for them. With basic calculations, I've found out that $M(1,k) = 1$, $M(2,k) = k+1$, $M(3,k)=\frac{1}{2} (k+1)(k+2)$ and $M(4,k) = \frac{1}{6}(k+1)(k^2+5k+6)$. I could not come up with an inductive proof for general $M(n,k)$, however.
And I'm not very interested in a proof, as I assume - maybe naively - that such a proof will be inductive, using only elementary calculations. I'm more interested in the result.
Do these numbers have a name? Is there a formula for $M(n,k)$?
Thank you for your help.