248 
ON A PROBLEM IN THE PARTITION OF NUMBERS. 
[204 
and we are thus led to the series 
1, 
1, 2, 
1, 2, 4, 6, 
1, 2, 4, 6, 10, 14, 20, 26, 
&c.; 
where, considering 0 as the first term of each series, the first differences of any series 
are the terms twice repeated of the next preceding series: thus the differences of the 
fourth series are 1, 1, 2, 2, 4, 4, 6, 6. It is moreover clear that the first half of 
each series is precisely the series which immediately precedes it; we need, in fact, 
only consider a single infinite series, 1, 2, 4, 6, &c. It is to be remarked, moreover, 
that in the column of totals, the total of any line is precisely the first number in the 
next succeeding line. 
Consider in general a series A, B, C, D, E, &c., and a series A', B', C, D', E', 
&c. derived from it as follows: 
A' = 1A, 
B' = 24, 
C' = 2A + B, 
D' = 2A + 2 B, 
E' = 2A + 2B + C, 
F' =2A+2B+ 2 G, 
&c.; 
viz. the first differences of the series 0, A', B', C', I)', E', &c. are A, A, B, B, C, G, 
&c. Then multiplying by 1, x, x\ &c. and adding, we have 
A' + B'x + G'x 2 4- &c. = (1 + %x + 2# 2 + ...) {A + Bx- 4- Gx 4 + &c.) 
= (A + Bx* + Cx* + &c.); 
and if we form in a similar manner A", B", C", D”, &c. from A', B', C', I)', Sac. and 
so on, we have 
A" + B"x + C"x* + &c. = (A' + B'x 2 + CV + &c.) 
1 — x 
1 4" X 1 4" X* / . rt /la n \ 
—— -—-—- (A 4- Bx 4 + Gx 8 + &c.), 
1 — X 1 — X 2 
and so on. Write 4 = 1, and suppose that the process is repeated an indefinite 
number of times, we have 
1 + + (&x 2 4- 4- &c. = 
1 4- x. 1 4- x 2 .1 4- x 4 . &c. 
1— X .1 — X* .1 — ¿P 4 . &c.