Let's define $f(n,k)$ as sum of expressions of $x_1 x_2 \dotsb x_k$ where summation is performed over all positive solutions for $n=x_1 + x_2 + \dots + x_k$ and i'm interested in closed form of $f(n,k)$
My idea was as follows: Let's design straightforward reccurence pattern $$f(n,k) = \sum_{i=1}^n i \times f(n-i,k-1) + [n=0 \wedge k=0]$$
if we say that $\mathcal{E}_k(x)$ is generating function for $f(n,k)$ (fixing $k$ here) then $$\mathcal{E}_{k}(x)=\frac{x}{(1-x)^2} \times \mathcal{E}_{k-1}(x) + 1$$ So it's convolution of $\langle i \rangle _ {i=1}^\infty$ and $\langle f(i,k-1) \rangle _{i=0}^\infty$ plus one. But i can't really move forward from there. Are generating functions good tools for this task? I'd appreciate some help on this.
If we can represent $n=x_1 + x_2 + \dots + x_k$ where $x_i$ is group of $x_i$ ones then the answer if number of such solutions with exactly one element picked out from each of $k$ groups. So the solution is little variation of stars and bars and the answer is $$\binom{n+k-1}{2k-1}$$