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 exjDression 



(1-1)"^ 



by giving to r its approjDriate 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 



(1-1)"" 



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

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



