Some Nice Sums Are Almost As Nice If You Turn Them Upside Down

Some Nice Sums Are Almost As Nice If You Turn Them Upside Down
By Moa Apagodu and Doron Zeilberger
Written: July 17, 2009.
[To appear in J. Combinatorics and Number Theory.]
.pdf .ps LaTeX source
The sum of all the binomial coefficients n!/(k!(n-k)!) is as nice as can be, namely 2ⁿ, but the sum of their reciprocals is also nice (but not quite as nice). Indeed in 1981, Rockett proved that the sum of the reciprocals is (n+1)/2ⁿ times Sum(2^j/(j+1),j=0..n).
But what about other classics? Like Chu-Vandermonde and Dixon? Here we show that the same phenomenon still holds, but some people may argue that the "almost nice" that we claim is pretty ugly. Well, beauty is in the eyes of the beholder, and at any rate we formally define what we mean by "almost nice", and by that definition, we are safely correct in our assertions.

Checking Maple Program
Important: This article is accompanied by a small Maple program

checkUpsideDown that empirically verifies the four identities.

Sample Input and Output for checkUpsideDown

The input yields the output

Doron Zeilberger's List of Papers
Doron Zeilberger's Home Page