The sum of the products vanishes for n = 1, 2, 3, 4, 5, but not for n>5; 
for 2 = 6, it is (—1)® (6!) = 720; and for n = 7, it is 
720(44+3+2+1+0—1— 2) = 5040. 
The third conclusion of (6) shows that if 
China + (a +d) -- (vst eds SIE? oi pane + (a + kd) 
and 
an) [oje*— [i] @ +a + (5) @+20'—... + Ht (A) @ + aah 
be multiplied term by term and the (/ + 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 (& + 1)? products be added; e.g. take k = 4,a = 1,d = 2 
(I) 1 ; 3 ‘ 5 ; 7 9 
(IT) fh ea , 654 oe hein il 94 
9725 Se ssa — Gees. 50040" = “O60 
Vit —-=4° 34 654 474 1°94 
1 iP een 3750 — 9604 6561 384 
3 35. == 979 11250" 28819 19683 1152 
5 5  —1620 18750 —48020 32805 1926 
7 7 -=9968 26250  —67228 45927 2688 
9 9 —2916 33750  —86436 59049 3456 
D5e S100 +93750 —240100  +164025 9600 
§2. 
The properties noted above, and many others, can be made to depend 
upon those of the sum 
k 
A Sc n=o (1 (F) Fh eR) eee 
i=0 
It is readily shown that 
(2) S(k, n) vanishes for k>n 
n 
(3) S(k, n) = —k > a S(k — 1, i—1) 
=k 
one Onan Se nae 
—_——_-—--_ 
