Puzzles | Statistical puzzles | Summing n

Given a target sum ${\displaystyle n}$, you can choose ${\displaystyle k}$ summands each of which is a number, ${\displaystyle n_{i}\in \{0,1,...\},i=1,...,k}$, such that ${\displaystyle \sum _{i=1}^{k}{n_{i}}=n}$. How many ways are there of doing this?

Here the notion of a sum is same as that of a permutation, so two sums are same ${\displaystyle iff}$ they contain the same summands in the same order. E.g. ${\displaystyle 2+3+1}$ and ${\displaystyle 1+3+2}$ are not the same.

While you are at it, whats the answer, if ${\displaystyle n_{i}\in \{1,2,...\}}$?

solution