2H EULER. 



This coefficient is 



n{n-l) ... {n-m-\- 2) n {n - 1) ... (n-m + 3) 



2 1 on - 1 ~ 4 I ?/z - 2 



n (n — 1) ... {n — m-\- 4) 

 ■^ 8 ! m - 3 



Then 8 — \m — 2 times this coefficient. 

 Hence with this expression for S we find that 



1 772 1 771 {m — 1) 



^i'n = T 



'" 4 7i - «i + 1 8 (n - ??i + 1) {n - ??z + 2) 



1 m {in — 1) (w — 2) 



IG (?i - wi + 1) {n - m + 2) (« - m + 3) 



+ 



. (_ 1 V^ -1 m(m-l)...2 

 ■^^ ^ 2"^ (yi-m + 1) ...(vi-1)' • 



This is the expression for the advantage of the Banker which 

 was given by Nicolas Bernoulli, and to which we have referred in 

 Art. 157. 



Now the form which Euler gives for w„ is 



m { m — 1 {m — l){m — 2){m — 2) 



2'" \ l{n-l) 1.2.3(/i-3) 



• • • ( • 



{m — 1) {m — 2) {m — 3) (m — 4) {m — 5) 

 + ^^ — ^ o A — ^7 ^^ i- 



1.2. 3. 4. 5(;z-5) 



Euler obtained this formula by trial from the cases in which 

 w = 2, 3, 4, . . . 8 ; but he gives no general demonstration. We will 

 deduce it from Nicolas Bernoulli's formula. 



By the theory of partial fractions we can decompose the 

 terms in Nicolas Bernoulli's formula, and thus obtain a series of 

 fractions having for denominators w — 1, w — 2, n — 3, . . . ?z — ?7i + 1 ; 

 and the numerators will be independent of n. 



We will find the numerator of the fraction whose denominator 

 is w — r. 



From the last term in Nicolas Bernoulli's formula we obtain 



{-ly^^ w(m-l)...2 



m — ^ — r ?' — 1 ' 



