433 
SOME PROPERTIES OF BINOMIAL COEFFICIENTS. 
A. M. Kenyon. 
81. 
The binominal coefficients of the expansion 
k\ k k\) k— k\ k-2 9 k) k 
(a + yy = (oz -b i}e 'y + 5a! aid > .B SE fi yh 
were known to possess a simple recursion formula 
; (k) ES a eta eel set 8) eae 
(1) eed ae ste fin = 0,1) 903" 
by means of which Paseal’s Triangle* 
) g 
FO te — Me en 2 tee ee) Gio 
- | 
kaa) 1 
eal 1 1 
k=2 1 2 1 | 
Se 1 3 3 1 | 
k=4 il 4 6 4 1 | 
etc = = = = = = | 
could be built up, before Newton showed that they are functions of k and n: 
(2) (| ae 0 eee hens Wench Sue aint 
t) nt ee ee 
k = ri | Des] . 
n| fell n=0 
A great number of relations involving binomial coefficients have been 
discovered**; some of the most useful of these are 
ee ees ha et | 
k 
Pid ile a wee, Obes 
*See Chrystal: Algebra I, p. 81. 
**See Chrystal: Algebra II, Chaps. XXIII, XXVII. Hagen: Synopsis der hoeheren 
Mathematik, p. 64; Paseal: Repertorium der hoeheren Mathematik I, Kap. II, See. 1. 
28—4966 
