My Contribution to the Mathematical Arts

Many, many years ago, sometime in high school, I learned that the formula for computing



is



I'm pretty sure we were shown the geometrical derivation of this formula. In college, in late 1975 or early '76, in a Math Lab one problem asked to guess the formula for



In my handwritten lab notebook I used induction to solve the specific case k=2:



However, I wanted to solve the problem once and for all for all values of k so that I would never have to do it again. I conjectured that the general solution would be of this form:



The derivation used induction on the general formula and found that the coefficients to the solution are:




Perhaps of interest are these three properties of the coefficients: