Creating a summatory list without iteration Let list $S_k$ be an arbitrary list of numbers (may not necessarily be ordered). List $S_{k+1}$ is created via the cumulative sum of elements from list $S_k$. For example if $S_k$ = [2,5,7,9] then $S_{k+1}$ = [2,7,14,23] Is there a way to tell what numbers will be in list $S_n$ with $n>k$ without needing to create all the intermediate lists?

Using $S_k(i)$ to indicate the $i^{th}$ term of $S_k$, then

$$S_n(j) = \sum_{i \le j} {n-k-1+j-i \choose j-i} S_k(i)$$ so you only need to do weighted sums over the original sequence.