Let $[n]=\{1,2,\ldots , n\}$. We are now going to pick an ascending sequence of length $r$ in $[n]$, i.e., $a_1,\ldots, a_r\in [n]$ such that $a_i\geq a_{i-1}$. The question is: in how many ways can I pick this sequence? Denote the answer by $s(r)$. Also, denote the number of such sequences that terminates with $k\in [n]$ with $s(r,k)$.
It is sort of a form of linear programming since we have $r-1$ linear constraints $a_i\geq a_{i-1}$. However, I find it difficult to count. We have $$ s(r)=\sum_{k\leq n} s(r,k)=\sum_{k\le n}\sum_{k_1\leq k} s(r-1,k_1)=\ldots\\ =\sum_{k\leq n}\sum_{k_1\leq k}\ldots \sum_{k_{r-1}\leq k_{r-2}} s(1,k_{r-1\\}) \\=\sum_{k\leq n}\sum_{k_1\leq k}\ldots \sum_{k_{r-1}\leq k_{r-2}} 1. $$
Now, we can, of course, sum those $\Sigma$'s one by one. using the formulae for $\sum n^k$, however, that would be very complicated.
Is there an easy way to count this number?
Notice that this is equivalent to counting the following $$(\alpha _1,\cdots, \alpha _n)$$ where $\alpha _i$ tells me how many times is $i$ in the sequence, and $\sum _{i=1}^n\alpha _i=r$ because there are $r$ elements in the sequence. This is called a composition of $r$ into $n$ parts and, by stars and bars, there are $$\binom{r+n-1}{n-1}.$$ Notice that using your approach you will have to use the HockeyStick sequentially, see that $1=\binom{k_{r-2}}{0}$ and each time you will add one and one.