289] ON SOME NUMERICAL EXPANSIONS. 471 
As an instance of the use of the preceding table, let it be required to find 
expressions for the combinations without repetitions of the series of natural numbers 
1, 2, 3,... (n— 1), or what is the same thing for the coefficients of the powers of k in 
k(k-l)(k-2)...(k-n + l). We have 
k(k-l)(k-2) ...(k-n + 1) 
= k n — A-P + A 2 n k ll ~ 2 — &c. 
= Un coefficient x n in (1 + x) k ; 
whence 
(—) r A r n = Un coefficient x n k n ~ r in (1 + x) k 
= Un coefficient x n k n ~ r in (1+iS) 
Yin 
= coefficient x n in {log (1 + x)) n ~ r 
= ff7 rTn —. coefficient af in j log ( 1+ ^)| n ~ r . 
11 (n — r) ( X J 
Thus 
A 1 n = n (?i — 1)}, 
AA = n{n-l){^(n- 2) 2 + (n - 2)}, 
A 3 n = n(n- 1) (n - 2) {Jg (n - 3) 3 + & (n - 3) 2 +£ (n - 3)}, 
and so on, as far as the expansion has been effected. It may be remarked that the 
general expression for the algebraical transcendent A r n is given in Dr Schlafli’s paper, 
“Sur les coefficients du développement du product (1 +æ)(1 + 2x) ... {1 +(?i — \)x) 
suivant les puissances ascendantes de x.” Crelle, t. xliii. [1852], pp. 1—22, but the law 
is a very complicated one. 
We have 
log (1 + x) = X — \x 2 + $X 3 — \x? + lx? — jrX 6 + lx 7 — &C. 
and dividing by x, and taking the logarithm, we have as before 
log (3 log (1 + x)) =-^x + fox 2 -K + + âWsïïïï« 6 - &c - 
which may be considered as the first of a series of logarithmic derivatives, viz., 
dividing by — \x and taking the logarithm we have 
log ( - \ log ^ log (1 + x)Jj = -fox + fo^x 2 - fo^fox? + ~ iæéWs æ5 + &c., 
and by the like process 
lo s(-i log (-I lo s(ï lo 8 O +*))))- 
— v2U æ + sifïïo^’ 2 — “b if 07ÏÏÏTÏÏ0 oïïïï — &C., 
and so on.