CHAP. Ill] PARTITIONS. 105 



so that the number of partitions of n into distinct integers equals the number 

 of partitions of n into odd parts not necessarily distinct. 

 Replace x by x 2 in (3) . Since n(l - z 2fc ) = PQ, 



Q = (1 - x 2 - x 4 + x 10 + x 14 - ) i 



i = 1 + x + 2x z + 3z 3 + 5z 4 + -. 



Hence, by multiplication, 



Q = i + x + x 1 + 2x 3 + 2z 4 + 3z 5 + 4z + . 



Thus the coefficient of x 3 in this series gives the number of partitions of s 

 into distinct parts. Since 



(1+ x)(l + z 2 )(l + x 4 ) - = 1 + x + x 2 + x 3 + -, 

 (x- 1 + 1 + z)(z- 3 + 1 + z 3 )(z~ 9 + 1 + x 9 ) . = 1 + x + x 2 + x 3 + 



+ or 1 + or 2 + or 3 + , 



every integer can be obtained by adding different terms of the progression 

 1, 2, 4, 8, 16, or of 1, db 3, 3 2 , -. The latter facts were known 

 by Leonardo Pisano, 11 Michael Stifel, 110 and FransvanSchooten, 12 who gave 

 a table expressing each number ^ 127 in terms of 1, 2, 4, , and every 

 number ^ 121 in terms of 1, 3, 9, 



Euler 13 reproduced essentially his preceding treatment. He concluded 

 ( 41, p. 91) that, if P(ri) or n (co) denotes the number of all partitions of n, 



P(n) = P(n - 1) + P(n - 2) - P(n - 5) *- P(n - 7) + P(n - 12) H ---- , 



the numbers subtracted from n being the exponents hi (3). His table of 

 the number n (n) of partitions of n into parts == m here extends to n =i 59, 

 m ^ 20 and includes m = <x> . He proved again that every integer equals 

 a sum of different terms of 1, 2, 4, 8, 



Euler 14 noted that the number (N, n, m) of partitions of N into n parts 

 each si m is the coefficient of X N in the expansion of (x + x 2 + + z m ) n . 

 Set 



(5) (1 + 3 + . . . -f. X m-l)n = I + AnX + BnX 2 + . . . f 



bring to a common denominator the derivatives of the logarithms of each 

 member and equate the coefficients of like powers of x in the expansions of 

 the numerators. The resulting linear relations determine A n , B n , in 

 turn, whence 



X(n + X, n, m} = (n + X l)(n + X 1, n, m) 



(mn + m X)(n + X m, n, m) 



+ (mn n + m + 1 X)(n + X m 1, w, w). 



11 Scritti L. Pisano, I, Liber abbaci, 1202 (revised about 1228), Rome, 1857, 297. 

 lla Die Coss Christoffs Rudolffs . . . durch Michael Stifel gebessert . . . , 1553. 



12 Exercitationum Math., 1657, 410-9. 



13 Novi Comm. Acad. Petrop., 3, ad annum 1750 et 1751, 1753, 125 (summary, pp. 15-18) ; 



Comm. Arith. Coll., I, 73-101. 



14 Novi Comm. Acad. Petrop., 14, I, 1769, 168; Comm. Arith. Coll., I, 391-400. 



