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Topic: Fun Proof by Induction
Replies: 0

 John Nahay Posts: 104 Registered: 12/6/04
Fun Proof by Induction
Posted: Jul 18, 1996 2:27 PM

My previous posting to this group has gone unanswered, no doubt, partly
due to the complexity of my setting it up. So, here is a special case of
the following.

Fix positive integer p >=2. Let G= power set of {1,2,...p-1}.
Let g be any element of G. Define g to be even if its size is even, odd
if its size is odd.

Let F(x,p) = sum over even g in G of product over i in G of (x+p-i)-choose-x

(The binomial coefficient).

Let H(x,p) = sum over odd g in G of product over i in G of (x+p-i)-choose-x.

Let D(x,p) = F(x,p) - H(x,p)

I have already verified for p=4 and p=5 what D(x,p) is. The obvious
conjecture derived from these cases is that

D(x,p) = (-1)^p/(p!) times Product of (x+2 -i) from i = 1 through p.

I have tried to prove this conjecture by induction but have not succeeded.