435 



The sum of the products vanishes for n = 1, 2, 3, 4, 5, but not for n>5; 

 for n = 6, it is (— l) 6 (6!) = 720; and for n = 7, it is 

 720(4 + 3 + 2 + 1 + — 1 — 2) = 5040. 



The third conclusion of (6) shows that if 



(I) a + (a + d) + (a + 2d) + + (a + M) 



and 



(II) (J) a* - [J] (a + tf) fc + g) (« + 2d) k - 4- (_i) fc [*] ( a + M) * 



be multiplied term by term and the (k + 1) products be added, the result will 

 be the same as though (II) be multiplied through by the terms of (I) in suc- 

 cession and the (k + l) 2 products be added; e.g. take k = 4, a = 1, d — 2 



(I) 



1 



3 



5 



, 7 



9 



(II) 



• l'l 4 



, -4' 3 4 



, 6'5 4 



, -4'7 4 



, 1 9 4 





1 



— 



972 + 



18750 - 



67228 



+ 5904J 



= 9600 





l'l 4 



— 4'3 4 



6'5 4 



— 4'7 4 





19 4 





1 



1 



- 324 



3750 



- 9604 





6561 



384 



3 



3 



- 972 



11250 



—28812 





19683 



1152 



5 



5 



—1620 



18750 



—48020 





32805 



192C 



7 



7 



—2268 



26250 



—67228 





45927 



2688 



9 



9 



—2916 



33750 



—86436 





59049 



3456 





25 



—8100 



+93750 



—240100 



+ 164025 



9600 



§2. 



The properties noted above, and many others, can be made to depend 

 upon those of the sum 



(1) S(k, n) =Z (— I] 



! = 



k) .n 



It is readily shown that 



(2) S(k, n) vanishes for k>n 



(3) S(k, n) = —k i \ n - Z !l S(k — 1, i — 1) 



i=k [ ' 



= - ~ i;!ij s(k-i,i-i) 



k, n = 0, 1, 2, 3, . . 



k, n =0, 1, 2, 3, 



