Al. a. Tchouproff 
143 
Putting 
1.2. 3. ..A 
i.e. dciiutiiig by (V the miiiiber of coiiibiniitioiis of k cleiuciits h at a time, we may 
express ak,k-n the form : 
'Xk,k-n='^ A,,,C^'^-^ (5), 
where the coefficients ^„,,- are independent o\ k and are detined by the rehitions : 
Hence 
.(6). 
J 
-4n,o — 1.3.5 . 
^„,j = 1.3.5. 
-4ji,2 = 1.3.5 . 
An,i =1.3.5. 
= 1.3.5. 
(2n-l) 
(2«-i)H«-]] 
(2n-3) {i[«-l]t-^'J + i["-lJ~''l 
(2» - 3) {J, - 2][-^] + 3^ - 2]f-J + ,L _ 2][-ij 
(2« - 5) Uh [" - 2]'-"^ + rk - 2]^-^J + nVir [" - 2]'-^ 
^„,, = 1 . 3 . 5 . . . (2// - 5) {^J,,^ - 3]t-TJ + _ + [„ _ 
The coefficient ^4 can easily be expressed in an independent form 
Putting 1 .2.3 ... t = i!, 
we shall have : 
.(7). 
.(8), 
where the siunmation extends to all possible positive integer values of i^, i.^, ... if, 
satisfying the relation : + 1.,+ ... + if= i, and to all integer values of Ji-^, li.,, ... /if, 
satisfying the conditions 
•2 ^ hi < I/., < ... < /if, 
/i 1 ii + /i.^L + ... + /ijij- = it + i. 
Introducing the notation 
and noting that 
we find from (5) : 
(9), 
