Let $A(z)=1+az+a^2z^2+\dots+a^nz^n$ and $B(z)=1+bz+b^2z^2+\dots+b^nz^n$, where $a,b \in \mathbb Z$. And let $C(z)=A(z)B(z)=1+c_1z+c_2z^2+\dots+c_{2n}z^{2n}$. Is it possible that $c_k=c^{k+1}$ for some $c \in \mathbb Z$ and some $k \in \{2,\dots,2n\}$? For which $k$?
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$c_2=a^2+ab+b^2$, so you want $a^2+ab+b^2=c^3$. There will be solutions if $c$ has no prime factor of the form $3m-1$. E.g., $18^2+(18)(1)+1^2=7^3$. |
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Other than the obvious $a=b=c=0$, I guess. Well, slightly less obvious is $a=b=c=k$ where $k \le n$. |
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