Let $k>0$ and $(l_1,\ldots,l_n)$ be given with $l_i>0$ and $\sum_{i=1}^n 2^{-l_i} > 1$(and the $l_i$âēs need not be distinct). How do I count the number of ordered sums that add up to $k$ that involve the $l_i$'s.
For example if $k=4$ and $l_1=2, l_2=2, l_3=4$, then $$k = l_1+l_1$$ $$k = l_1+l_2$$ $$k = l_2+l_1$$ $$k = l_2+l_2$$ $$k = l_3$$ And so there are $5$ ways. This is not really a composition of the integer since $l_1$ and $l_2$ are both $2$ but considered distinct.
If there is not a reasonably simple expression, is there some known asymptotic behavior as a function of $k$ with $(l_1,\ldots,l_n)$ fixed? In particular I would like to know if the condition $\sum_{i=1}^n 2^{-l_i} > 1$ guarantees that the growth will be like $a^k$ where $a>2$.