TITLE: What is this polynomial zero everywhere but $n!$ at point $n$?
QUESTION [1 upvotes]: For an integer $n \geq 2$, define a set of $n$ points $(p_{x,1},p_{y,1}),\ldots,(p_{x,n},p_{y,n})$ as follows. For $1 \leq i \leq n-1$, let $(p_{x,i},p_{y,i}) = (i,0)$, and $(p_{x,n},p_{y,n}) = (n,n!)$. Put differently, the first $n-1$ points evaluate to zero, and the last point to the factorial of $n$.
For any $n \geq 2$, take the interpolating polynomial $P_n$ through the point set defined above. For example, for $2 \leq n \leq 4$, we get:

$P_2 = x$,
$P_3 = -x+x^2$, and
$P_4 = 2 x - 3 x^2 + x^3$.

I can keep going with the help of a computer easily, but how can you generalize this for any $n$? Is there some nice closed form (and how to find it)? This is probably easy, I just haven't worked with polynomials in a long time. Maybe this is even some known polynomial that has a name.

REPLY [4 votes]: Another way to write polynomials is as a product, not a sum.
For example, your $P_4=x(x-1)(x-2)$.  That way, it is clear that $P_4=0$ when $x=0,1,2$, and $P_4(3)=6=3!$.
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