TITLE: Is this similar to a known combinatorial identity?
QUESTION [18 upvotes]: (Apologies if this is  too obscure.)
In joint work with Izzet Coskun we came across the following kind of combinatorial identity, but we weren't able to prove it, or to identify what kind of identity it is. (We looked in some references, but to the outsider it can be difficult to distinguish one insanely complicated sum of binomial coeffficients from another...)
The identity looks like the following. Say $n$ is a fixed natural number, and $i \leq n$ is an even natural number. (There is an analogous formula when $i$ is odd.) Then
$\begin{align}
\sum_{m=2}^{\frac{i}{2}} (x-1) \left(x-2\right)^{2m-3} \ \sum_{j=0}^{i-2m} \binom{n-1-j}{i-2m-j} 2^{j+2} \sum_{l=0}^j (-1)^l \binom{n+1}{l} \, x^{i-2m-l} & \\\\
+ \sum_{j=0}^{i-2} \binom{n-1-j}{i-2-j} 2^{j+1} \sum_{l=0}^j (-1)^l \binom{n+1}{l} \, x^{i-2-l} &\\\\ 
+ \sum_{j=0}^{i-2} \binom{n-1-j}{i-2-j} 2^{j-1} \sum_{l=0}^j (-1)^l \binom{n+1}{l} (i-2-l) \, x^{i-l-1} & \\\\
- \sum_{j=0}^{i} \binom{n-1-j}{i-j} 2^{j-1} \sum_{l=0}^j (-1)^l \binom{n+1}{l} (i-l) \, x^{i-l-1} & \\\\
\qquad =C_i \left(x-2 \right)^{i-1} 
\end{align}
$
where $C_i$ is some constant that can be read off fairly easily by looking at the coefficient of $x^{i-1}$ on the left-hand side. (There are various rewritings one could do to the left-hand side, but it's not clear how much this helps.)
Given that there are various irregularities in the last three terms on the left-hand side, it seems unlikely that the whole thing can be reduced to a simple form. But it would nevertheless be very useful to know if this looks similar to any known combinatorial identities.
Maybe a simpler warmup question would be: how to see that the sum of the last three terms on the left-hand side is divisible by $x-2$? That might help get going with an inductive argument. 
Any ideas would be much appreciated!

REPLY [3 votes]: (For completeness, let me write as an answer the comment I made on Zack Wolske's answer.)
It turns out that this identity is tractable using the Wilf-Zeilberger machinery, as suggested by David Lehavi in comments to the question. Christoph Koutschan was able to prove the identity
using the Mathematica package $\mathtt{HolonomicFunctions}$ which he developed in his thesis:
Advanced Applications of the Holonomic Systems Approach (PhD Thesis). RISC-Linz, Johannes Kepler University, September 2009. 
Unfortunately I don't understand enough about the method to describe it accurately here, but very roughly, the idea is to prove identities of this kind by calculating the ideal of recurrences and differential operators that each one satisfies, and then checking that a certain (explicitly computable!) number of initial values agree.  Follow the above link for a real explanation!