456 Proceedings of Royal Society of Edinburgh. [sess. 
for, the successive results being equal to 
i (i+i), 
i(i+i)(i+i+i), 
i (i+i)(i+i+i)(i+i+i+i), 
are equal to 
2! , 3! , 4! , . . 
(20) It is only a variant of the statement in § 18 to say that the 
numbers of the 2nd, 3rd, 4th, .... columns of the table are the 
coefficients in the expansions of 
1 + x 
(1 +a?)(l +x + x 2 ) 
(1 +#)(1 +x + 0(1 +x + x 2 + x 3 ) 
respectively. Consequently we have the result 
V n s = coefficient of x s in the expansion of 
(1 + #)(1 + X + x 2 )(l + X + X 2 + X 3 ) . . . (1 + X + X 2 + ... + x n ~ l ) 
or of 
(1 - x)(l - x 2 )(l -X s ) (1 - 0/(1 - x) n . 
Taking the case of n = 4 as an example, we have 
(1 - £e)(l ~ 0(1 “ 0(1 “ 0‘(1 “ 
= (1 - x-x 2 + 2x 5 - x s - x 2 + £ 10 ) 
•(1 + C 4il a? + C 5i2 ^ 2 + C 6>3 ^ 3 + ) 
1 
= v 4 , 0 
+ (C 4.1-1)* 
+ V 4.1'* 
+ ( 0 M -C«- 1 )^ 
+ V4.2 -* 1 
+ (Q 6,3 — ^5.2 — Wl )^ 3 
+ V 4 . 3 - 
+ (C 7,4 “ C^^ - C 5>2 )^ 
+ . . . . 
+ (C 8 ,5-C7,4-C 6>3 + 20 
+ (C 9,6-C 8 ,5-C7,4+2C 4>1 )^ 
+ (^io . 7 ~ C 9>6 - C 8 , 5 + 2 C 5 > 2 )x 7 
so that 
^4.S = ^3+S,3 ” ®2+S.3 “ ^1+ .3 + ^^5 - 2,3* 
