Engler — Figurate Numbers. 



45 



in which the diagonal indicated ascending from left to 

 right gives the coefficients of the ordinary binomial expan- 

 sion in terms of r. 



Formnla (7) is valid for negative as well as positive 

 values of v, in fact, for all integral values of r from 

 — 00 to +00. 



For values of r from to — oo we obtain the different 

 series indicated in Table 10^ which is simply an extension 

 of Table 6 upwards. 



Table 10. 



Ail the properties which have been described above as 

 applicable to the different series with positive indices 

 belong also to the portion of Table 10 with negative 

 indices, as will be at once seen ]3y trial. 



* In Tables 10, 11, 12 and the following, negative values are indicated for 

 convenience by minus signs placed above tlic numbers. 



