Is there a closed-form solution to this recurrence?

A friend and I were examining polynomials of the form $p_n (x) = x (x+1) (x+2) \cdots (x+n -1)$ and we were trying to come up with some kind of closed form for the coefficients when the polynomial is expanded. We came up with the recurrence:

$$c_{n,k} = (n-1)c_{n-1,k-1} + c_{n-1,k}$$

Where $c_{n,k}$ denotes the $k^{th}$ coefficient of $p_n$ (that is, $p_n(x) = c_{n,0}x^n + c_{n,1}x^{n-1} + \cdots c_{n,n}$). We spent a good 9-10 hours working on the recursion, and got a few properties, but couldn't seem to break it down any further. Here's what we got:

$$p_n (1) = n!\\ \Rightarrow \sum_{k=0}^n c_{n,k} = n!\\ c_{n,n-1}=(n-1)!\\ \Rightarrow \sum_{k=1}^{n-2}c_{n,k} = n!-(n-1)!\\ c_{n,n}=0\\ c_{n,1}=1\\ c_{n,2}=\frac{n(n-1)}{2}\\ c_{n,3} = \frac{1}{24} n(n-1)(n-2)(3n-1)$$ The last two are just brute-force calculations to see if we could find a pattern, but after three it became arduous. I threw the recurrence into Mathematica and couldn't get any results. Any ideas on this?

[Edit: wrote the polynomial incorrectly as pointed out in the comments.]

• Vieta Feb 23, 2014 at 6:10
• Your recursion seems to be using $p_n(x) = c_{n,n} + c_{n,n-1} x + \ldots + c_{n,0} x^n$ Feb 23, 2014 at 6:56

$$p_n(x) = \sum_{k=0}^n \left[{n\atop k}\right] x^k,$$
where $\left[{n\atop k}\right]$ are the Stirling numbers of the first kind. So, you can see that the coefficients are given by the Stirling numbers of the first kind.
Your polynomial $p_n(x)$ is a Pochhammer's polynomial. It can be expresses as $\Gamma(x+n)/\Gamma(x)$ or with several other relationships that you will find in : http://mathworld.wolfram.com/PochhammerSymbol.html