TITLE: Identity with binomial coefficients and k^k
QUESTION [13 upvotes]: In process of doing some computations on Hilbert schemes, I stumbled across the following identity, for $k \ge 2$:
$$
k^{k-3} = \frac{1}{2} \sum_{i=1}^{k-1} \binom{k-2}{i-1} i^{i-2} (k-i)^{k-i-2} 
$$
which can be verified computationally for small values of $k$ (I did it for the first 500).
This is actually a special case of a bigger problem, so I'm hoping to get ideas about what methods might be applicable to prove this kind of identity. 
I understand that there are good methods for automatically proving hypergeometric identities, but am I correct in thinking that this is not hypergeometric, due to the $k^{k-3}$ type terms?
EDIT
Here is a link to the bigger question that I had that this is a special case of: Counting some binary trees with lots of extra stucture

REPLY [2 votes]: In my (first research) paper, A note on evaluation of Abel's sums, (that appeared second, in 1979) I described another way to evaluate Abel's sum of the form $$A_n(x,y;p,q)=\sum_{k=0}^n {{n} \choose {k}}(k+x)^{k+p}(n-k+y)^{n-k+q},$$ via (simpler) auxiliary sums defined by $$F_n(x,y;p,q)=\sum_{k=0}^n {{n} \choose {k}}(-1)^k (k+x)^{p}(n-k+y)^{q}.$$ The $F_n$ functions have simple evaluations in terms of difference operators. I was inspired by Riordan's book Combinatorial Identities and later corresponded with Riordan, and met him in 78.