Given $y_{n}(x)=\left(\sum_{k=0}^{n-1} x^{2^k}\right)^{n}$. An example ($n=5$) may look like $$ y_5(x)={x^{80}}+5 x^{72}+5 x^{68}+5 x^{66}+5 x^{65}+10 x^{64}+20 x^{60}+20 x^{58}+20 x^{57}+20 x^{56}+20 x^{54}+20 x^{53}+40 x^{52}+20 x^{51}+40 x^{50}+30 x^{49}+35 x^{48}+60 x^{46}+60 x^{45}+60 x^{44}+60 x^{43}+80 x^{42}+50 x^{41}+61 x^{40}+60 x^{39}+100 x^{38}+90 x^{37}+85 x^{36}+70 x^{35}+95 x^{34}+65 x^{33}+75 x^{32}+{120 x^{31}}+120 x^{30}+100 x^{29}+110 x^{28}+100 x^{27}+90 x^{26}+90 x^{25}+100 x^{24}+100 x^{23}+90 x^{22}+70 x^{21}+66 x^{20}+70 x^{19}+55 x^{18}+65 x^{17}+75 x^{16}+60 x^{15}+50 x^{14}+50 x^{13}+40 x^{12}+30 x^{11}+31 x^{10}+25 x^{9}+15 x^{8}+10 x^{7}+5 x^{6}+x^{5} $$ If you plot exponents of addends of $y_n$ against prefactors, this looks (for $n=7$) like
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Can this be described by a kind of Poisson distribution for general $n$?
This is not an answer, but a series of thoughts about the problem.
As a start, the coefficient of $x^k$ in $y_n(x)$ seems to be the number of ways that $k$ can be written as the sum of $n$ powers of $2$ not exceeding $2^{n-1}$.
Another approach might be to use the fact that the coefficient of $x^m$ in $f(x)$ is $\frac{f^{m}(0)}{m!}$, and evaluate $y^{m}(0) $.
Another thought is that having $n$ as both the exponent and limit of the sum seems odd. It might be profitable to consider $y_{n, m}(x) = \left(\sum_{i=0}^n x^{2^i}\right)^m $.
That's all I can think of for now.