Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
How can we prove $4^{-n}(n+1)\left(2^{2n+1}-{2n+1\choose n+1}\right)$ produces the central term in the $2n+1$th row of the following pyramid? The pyramid is produced in this way: atop the triangle is 1 and each number below it is determined by summing half of each of the numbers that it is supporting, and adding 1 to the sum.
1 3/2 3/2 7/4 10/4 7/4 15/8 25/8 25/8 15/8 31/16 56/16 66/16 56/16 31/16
I have attempted induction and it does not work well as I am sure you can see. I do not know how to prove the formula... even an intuitive proof would work.
Please see my comment below about constructing the pyramid as a whole using coefficients. Another question to all(ESP joriki) : what can be said in general about any term in the pyramid? Can we make a generalization of any term in the pyramid based on The row number and the $k+1$th or kth term in a row, perhaps, depending on how you define k? I have been looking at it and it seems that writing it in terms of k and z, where z=n-1 might make it simple, but I don't know.