First: We (Bill Gasarch, Clyde Kruskal, Justin Kruskal) have in a paper of ours a summation that we proved. Is it new? Here is the summation: (It uses a sum that I previously inquired about here and got useful information, including that it was originally proved by Euler.)
Let p(x) &isin Z[x] be a polynomial of degree n with constant term 0.Our proof is here.
p(s) - &sumk=1n(s+k-1 choose k)&sumi=0k-1(k-1 choose i)(-1)i (p(s+i+1) - p(s+i)) = 0
(Side Request: How do you do a choose-sign in html?)
Second: I am, as always, tracking down lower bounds on the VDW numbers. I need the article An application of the Lovasz Local Lemma, a new lower bound for the Van der Warden numbers by Zoltan Szabo, Random Structures and Algorithms, 1990. Electronic preferred, but will take what you got. SO, if someone has it please email me article or pointer or whatnot. (Pointer might not work if your library has access to this journal, since ours does not have it in electronic or hardcopy.)