. Carnegie Institution of Washington publication. CHAP, ill] PARTITIONS. 127 is (N2 - l)/8 or (N + 2)(N + 4)/8 according as N is odd or even. An anonymous writer (pp. 93-4) stated that the number of sets of positive integral solutions of x{ + • • • + xm = N is {N} — m{(N — j)/2], where {i} = ('""'"i"1) and j = 1 or 2 according as N is odd or even. K. Weihrauch74 discussed the number fn(A) of sets of solutions of ... + anxn = A, where the a's are positive integers. Set P = aia2 • • -an, Si = a{+ ••• + a*n, A = pP + m, where m is one of the integers 1, • • •, P. Then = p. fn(
. Carnegie Institution of Washington publication. CHAP, ill] PARTITIONS. 127 is (N2 - l)/8 or (N + 2)(N + 4)/8 according as N is odd or even. An anonymous writer (pp. 93-4) stated that the number of sets of positive integral solutions of x{ + • • • + xm = N is {N} — m{(N — j)/2], where {i} = ('""'"i"1) and j = 1 or 2 according as N is odd or even. K. Weihrauch74 discussed the number fn(A) of sets of solutions of ... + anxn = A, where the a's are positive integers. Set P = aia2 • • -an, Si = a{+ ••• + a*n, A = pP + m, where m is one of the integers 1, • • •, P. Then = p. fn(A) = fn(m) (r-2q)V the last being stated without proof, where e is the largest integer ^ r/2, R = m — Si/2, and = Z -, I ^ • • • (2a +4/3 + 67 + - - • = 2s), , /s, ...a " ' the 5's being Bernouilli numbers. Cf. Meissel135 and * E. Meissel75 treated the partition of very large numbers. E. Lemoine76 noted that every power n" of an integer n equals a sum of nk consecutive terms from 1, 3, 5, 7, • • -, if /i ^ 2k. Cf. Fre" G. B. Marsano's77 Table 1 is an extension of Euler's table of partitions of n into m parts, for n si 103, m ^= 102. Table 2 gives the coefficients as far as £53 of the expansions of St =' •"' »' s " n (1 " XJ]~1' and the coefficients as far as x107 of the expansion .of the first ten functions. The results for S/(l — x) give the number of ways of partitioning a number into parts 1, 1', 2, 3, • • -. Those for S/(l - x)(l - z2), into parts 1, 1', 2, 2', 3, 4, .... 74 Untersuchungen Gl. 1 Gr., Diss. Dorpat, 1869, 25-43. Zeitschrift Math. Phys., 20, 1875, 97, 112, 314; ibid., 22, 1877, 234 (re = 4); 32, 1887, 1-21. 75 Notiz iiber die Anzahl aller Zerlegungen sehr grosser ganzer positiver Zahlen in Summen ganzer positiver Zahlen, Progr., Iserlohn, 1870. 76 Nouv. Ann. Math., (2), 9, 1870, 368-9; de Montferrier, Jour, de math. e"lem., 1877, 253. 77 Sulla legge delle d
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