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 show that the solution to (3) is:



Having solved this specific case, I wanted to see if there was a general solution for any positive integer n and any positive integer exponent. Based on equations (2) and (4), I conjectured that the general solution would be a polynomial of this form:



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




Computed coefficients for 0 <= k <= 10 are
here.

Perhaps of interest are these properties of the coefficients:










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