<s>[INST] The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})?$ What is the coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\cdots (1+x^{100})?$

I manually expanded $(1+x)(1+x^2)(1+x^3)...(1+x^{10})$ and calculated the coefficient of $x^9$ as $8$ but i dont know how to solve it without expanding.Please help me.
 [/INST] The coefficient of $x^9$ is the number of partitions of $9$ into distinct parts, i.e., the number of ways of writing $9$ as the sum of distinct positive integers when the order of the summands doesn’t matter. The sequence of these numbers is OEIS A000009, and as you can see there, there is no nice formula. However, for small $n$ it’s not hard to write out the partitions by hand: $9$, $8+1$, $7+2$, $6+3$, $5+4$, $6+2+1$, $5+3+1$, and $4+3+2$.
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