<s>[INST] How to find the coefficient of $x^{203}$ in the expansion of $(x-1)(x^2 - 2)(x^3-3)\dots(x^{20} - 20)$? 
How to find the coefficient of $x^{203}$ in the expansion of $(x-1)(x^2 - 2)(x^3-3)\dots(x^{20} - 20)$?

I took $x$ as common from each bracket making $x^{190}$ but I don't understand what to do next. Please help.
 [/INST] Hint. Note that $\sum_{k=1}^{20} k=210$ and
$$(x-1)(x^2 - 2)(x^3-3)\cdots(x^{20} - 20)=x^{210}
\left(1-\frac{1}{x}\right)
\left(1-\frac{2}{x^2}\right)
\left(1-\frac{3}{x^3}\right)\cdots\left(1-\frac{20}{x^{20}}\right).$$
Now consider the integer partitions with distinct parts of $210-203=7$:
$$7,\quad 6+1,\quad 5+2,\quad 4+3,\quad 4+2+1.$$
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