204] 
ON A PROBLEM IN THE PARTITION OF NUMBERS. 
249 
and the coefficients 1, 58, (£, ( X), &c. are precisely those of the infinite series 1, 2, 4, 6, 
&c. We have more simply 
1 
+ 23.« + (£« 2 + 2)« 3 + &c. = 
1 
(1 — X) 2 (1 — X?) (1 — X*) (1 — X 8 ) & c. ’ 
which gives rise to the following very simple algorithm for the calculation of the 
coefficients : 
1, 
2, 
3, 
4, 
5, 
6, 
7, 
8, 
9, 
10, 
11, 
12, 
13, 
14, 
15, 
16 
0, 
0; 
1, 
2, 
4, 
6, 
9, 
12, 
16, 
20, 
25, 
30, 
36, 
42, 
49, 
56 
1, 
2, 
4, 
6, 
9, 
12, 
16, 
20, 
25, 
30, 
36, 
42, 
49, 
56, 
64, 
72 
0, 
o, 
o, 
0; 
1, 
2, 
4, 
6, 
10, 
14, 
20, 
26, 
35, 
44, 
56, 
68 
1, 
2, 
4, 
6, 
10, 
14, 
20, 
26, 
35, 
44, 
56, 
68, 
84, 
100, 
120, 
140 
o, 
0, 
o, 
o, 
o, 
0, 
o, 
0; 
1, 
2, 
4, 
6, 
10, 
14, 
20, 
26 
11 
2 | 
4, 
6 1 
10, 
14, 
20, 
26 | 
36, 
46, 
60, 
74, 
94, 
114, 
140, 
166| 
&c. 
The last line is marked off into periods of (reckoning from the beginning) 1, 2, 4, 8, 
&c.; and by what has preceded, the series which gives the number of 1-partitions, 
2-partitions, 3-partitions, &c. is found by summing to the end of each period and 
doubling the results; we thus, in fact, obtain (1), 2, 6, 26, 166, 1626, &c.: and the 
same series is also given by means of the last terms of the several periods. 
The preceding expression for 1 + 58« + Gt« 2 + &c. shows that 58, 6, &c. are the 
number of partitions of 1, 2, 3, 4, 5, 6, &c. respectively into the parts 1, 1', 2, 4, 8, 
&c.: and we are thus led to— 
Theorem. The number of «-partitions (first part unity, no part greater than twice 
the preceding one) is equal to the number of partitions of 2 a; ~ 1 — 1 into the parts 
1, T, 2, 4,... 2 a:-2 . Or, again, it is equal to twice the sum of the number of partitions 
of 0, 1, 2,... 2 X ~~— 1 respectively into the parts 1, 1', 2, 4,... 2 X ~ 3 (where the number of 
partitions of 0 counts for 1). 
For example, the partitions of 0, 1, 2, 3, &c. with the parts 1, 1', 2,... are 
(•) 
1, V, 
1+1, 1 + 1', l'+l', 2, 
1+1 + 1, 1 + 1 + 1', l + l' + l', l' + l'+l', 2 + 1, 2+r, 
the numbers of which are 1, 2, 4, 6. Hence, by the first part of the theorem, the 
number of 3-partitions is 6, and by the second part of the theorem, the number of 
4-partitions is 
2 (1 + 2 + 4 + 6), = 26. 
2, Stone Buildings, March 17, 1857. 
C. III. 
32