An Alternating Direction Method

combination of a's yields the numerical coefficient for that combination. For example, in the summation for Su given in Table 2, 1/ao4 has as the ai*-1aB_*+i coefficient ai3a8 which, in turn, has the numerical coefficient 11. The ai2a3a6 combination has 2 ! X 1 ! X 1 ! as the product of the factorials of the exponents. If 11 X 3 ! X 1 ! is divided by 2 ! X 1 ! X 1 !, the result is 33 which is the numerical coefficient of the ai2a3ae combination. Construction of Table 1. Table 1 shows the number of coefficients of — l/a0 as the extreme right-hand entry corresponding to a given n. The next entry to the left is the number of coefficients for -rT/a02, etc. The pattern for the number of — l/a0 and + l/a02 coefficients is obvious. It should also be noted that entries to the left of the stepped division remain fixed for n and k. The entries in a column for k = (n — h), where h = 0, 1, 2, etc., are formed from cumulative sums, starting from the right, of the entries in the row for n = h. For example, for n = 6, h = 5, and k = 1, the entry is 1. This entry is equal to the first 1 on the right in the row for n = 5. For » = 7, h = 5, and k = 2, the entry is 3. This entry is equal to (1 + 2) which is the sum of the first two right-hand entries of the row for n = 5. A column is formed in this fashion until the sum of the entries in the row (in this case, the fifth) is arrived at, after which the entries in the column remain fixed at this value. The fixed values are those to the left of the stepped division, as explained above. Summary. The method of finding Sn may be summarized briefly as follows. First, obtain the number of coefficients of each (— l)*/ao* Irom Table 1 or an extension thereof. Second, by some systematic procedure obtain as many different combinations of the a's as there are coefficients specified by Table 1, making certain that the weight of each combination equals n and the degree equals k. Third, find n X (k — 1) ! divided by the product of the factorials of the exponents to obtain the numerical value of a combination. A listing of Sn from » = 1 to n = 11 is given in Table 2.