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I was actually learning about this "pentagonal number" theorem. In the book it's mentioned that,

$$(1-x-x^2+x^5+x^7-x^{12}-x^{15}+\cdots)(1+p_1x_1+p_2x^2+p_3x^3+\cdots)=1$$

an then, all of a sudden, it says that the coefficient of $x^n$ in the product is zero, and therefore --

$p_n - p_{n-1}-p_{n-2}+p_{n-5}+p_{n-7}-p_{n-12}-p_{n-15}+\cdots = 0$

I do not quite understand how do I get to line 2 from line 1 above?

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    $\begingroup$ Try pairing one from the left and one from the right: $(1, x^n)$, $(-x, x^{n-1})$, $\ldots$. $\endgroup$ Sep 23, 2016 at 17:34

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