# What is $\prod_{k=1}^n (1-x^k)$?

I'd like to know what $$\prod_{k=1}^n (1-x^k)$$

evaluates to (assuming there is a simple closed form) and what it "is" in the context of commutative algebra (of which I knew little and recall less).

I'm sure I've seen this in the past but don't know where to place it. LaTeX search doesn't help.

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That's probably the simplest form. It's zero on roots of unity up to degree $n$. – knucklebumpler Mar 30 '11 at 21:39
What does "what it 'is' in the context of commutative algebra" mean? It is a certain polynomial. I don't know what else there is to say. What do you want to know about it? – Qiaochu Yuan Mar 30 '11 at 21:40
Looks like a finite mathworld.wolfram.com/q-PochhammerSymbol.html – deoxygerbe Mar 30 '11 at 21:42
If it's got a name (e.g., "the Herp-Derp polynomial"), or other stuff that will help me find context for it online. – S Huntsman Mar 30 '11 at 21:43
@Qiaochu: Assuming it's hard to evaluate in general, are there special values of $n$ for which it's easy to evaluate? – S Huntsman Mar 30 '11 at 21:53

Well, one has $$\prod_{n\geq1}(1-x^k) = \sum_{-\infty\leq n\leq\infty}(-1)^nx^{(3n^2-n)/2}.$$ This is a consequence of Jacobi's triple product identity.
Right, but the OP's question is about the finite product. I guess this tells us what the first $n$ coefficients are. – Qiaochu Yuan Mar 30 '11 at 21:51