Notation for a sum product

Question

I'm struggling with notation for a sum product. Let $f:Z^+\rightarrow Z^+$. I am interested in a sum where each term is the product of functions whose sum of arguments equals $n$. For example if $n=3$ then the sum product is $$ f(3)+f(2)f(1)+f(1)f(2)+f(1)f(1)f(1) $$ I would like to have something that looks like $$ \sum_{?} \prod_{?} (f(j))^? $$ and I'm most interested in the product. The closest I get is $$ \sum_{\Large \begin{array}{c} i_1,\ldots,i_d \\ i_1+\ldots+i_d=n \end{array}}f(i_1)\cdots f(i_d) $$ Is it possible to go further? Is there an alternative way to express this?