Engler — Figurate Numbers. 47 



The relation of Tables 10 and 11 to each other is shown 

 by their combination in Table 12. 



Table 12. 



It is interesting to note that every one of the series in 

 Table 10 represents the expansion of the expression 



(1-1)-'" 



by giving to r its appropriate value as shown by the index 

 of the series, that is, by the number of the row, for nega- 

 tive as well as positive values ; and that similarly every 

 one of the series in Table 11, reading from the top down- 

 wards, represents the expansion of the expression 



(1-lP 



by giving to n its appropriate value as shown by the number 

 of the column, for negative as well as positive values. 



