Suppose I have the following polynomial, $$f(x)=(1+x)^2(1+x+x^2+x^3)^2$$ expanding this gives: $$f(x)=1+4x+8x^2+12x^3+14x^4+12x^5+8x^6+4x^7+x^8$$ now suppose I want to extend this as follow: $$f(x)=(1+x)^2(1+x+x^2+\cdots+x^n)^{n-1}$$ where $n$ is a positive integers and can be odd/even, I wonder how to write the coefficients analytically in terms of $n$?

  • 5
    $\begingroup$ you have the geometric series in the second bracket $\endgroup$
    – Milan
    Commented Jun 3, 2019 at 17:03
  • 1
    $\begingroup$ Writing coefficients in terms of $n$ and expressing $f(x)$ in closed form are two different things. The comment above (+1) does the latter, not the former. $\endgroup$
    – Macavity
    Commented Jun 3, 2019 at 17:08
  • $\begingroup$ @Macavity indeed I need the coefficients in term of $n$. $\endgroup$
    – Wiliam
    Commented Jun 4, 2019 at 10:36
  • $\begingroup$ I have edited the question. $\endgroup$
    – Wiliam
    Commented Jun 4, 2019 at 10:37


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