Vol. 5 No 2 - Pi Mu Epsilon

k 1k - q + l ' kThus(lo) %+l,q+l =That is, (7) holds also for the (q+l)th element (counting from thetop down) of the column headed by R 1This concludes the Second-Principle Induction, and thereforealso the proof, that (7) holds for every element of the column headedSince m was an arbitrary integer greater than 1, we seeby RmSl .that (7) holds for every element of the coefficient triangle not inthe rightmost column. THEOREM 1 is therefore proved.The validity of our SECOND ALGORITHM for the determination ofthe coefficient triangle is now established. We describe this algorithm:IUse (4) to construct the first upper-right-to lower-left diagonal:.I1I11Determine R j,jl for each positive integer j. in either of thefollowing two ways:a) Apply the FIRST ALGORITHM to only the right-most column;b) When the jth row is known except for the element R j,j'use equation ( 2) ;Use equation (7) to determine R r.S for every r and s in I suchthat 2 < s