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I wonder if there exists a formula for this function? $$f(x) = (x-1)(x-2)(x-3) \cdots (x-k)$$ I want to know the coefficient of each $x^i$, and the first thing I came up with was to find the expansion of this expression. Any idea?


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I'm guessing that the $k$ in the formula for $f$ is not the same as the $k$ in "I want to know the coefficient of each $x^k$", so I suggest you change the latter to "of each $x^i$" to avoid confusion. – Arturo Magidin Apr 15 '11 at 17:23
@Arturo Magidin: You're just so right ^_^! Thank you. – Chan Apr 15 '11 at 17:23
up vote 11 down vote accepted

Look up the Stirling numbers and the falling factorial.

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Alternatively, one can use Pochhammer symbols to represent these entities. – J. M. Apr 16 '11 at 18:03

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