346 Mr. A. Cayley on a Problem in the Partition of Numbers. 



and if we were to proceed to the 4-partitions, each 3-partition 

 ending in 1 would give rise to two such partitions ; each 3-par- 

 tition ending in 2 to four such partitions ; each 3-partition end- 

 ing in 3 to six such partitions ; and each 3-partition ending in 

 4 to eight such partitions. We form in this manner the Table — 



Number of 



1-partitions. 



2-partitions. 



3-partitions. 



4-partitions. 



5 -partitions. 26 



&c. 



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 as the first term of each series, the first 

 difi"erences of any series are the terms twice repeated of the next 

 preceding series : thus the diflFereuces of the fourth series are 

 1, 1, 2, 2, 4, 4, 6, 6. It is moreover clear that the first half 

 of each sei'ies is precisely the series which immedia,tely 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'=2A 

 C'=2A-f-B 

 D'=2A-<-2B 



E'=2A-|-2B + C 



F=2A-f2B + 2C 



&c.; 



viz. the first difi'erences of the series 0, A', B', C, D', E', &c. are 

 A, A, B, B, C, C, &c. Then multiplying by 1, x, x^, &c. and 

 adding, we have 



