EULER. 2i3 



436. The next memoir by Euler is entitled Sur Vavantage du 

 Banqiiier cm jeu de Fharaon. This memoir was published in the 

 volume for 1764 of the Histoire de V Acad.... Berlin; the date of 

 publication is 1766 : the memoir occupies pages 144 — 164. 



437. Euler merely solves the same problem as had been 

 solved by Montmort and Nicolas Bernoulli, but he makes no refer- 

 ence to them or any other writer. He gives a new form hoAvever 

 to the result which we will notice. 



Consider the equation in Finite Differences, 



m {in — 1) (;? — m) (n — ??2 — 1 ) 

 ^'" " 2n (ii - 1) ^ 71 (n - 1) ""-' • 



By successive substitution we obtain 



m (m — 1)S 



u„ = 



" 2n{?i-l){n-2) ... (n-m + 1)' 



where S denotes the sum (f) (u) + </> (?i — 2) + (^ (/i — 4) + . . . , 



(f> (>i) being (ii — 2) (n — S) ... (n — m + 1). 



This coincides with what we have given in Art. 155, supposing 

 that for A we put unity. 



AYe shall first find a convenient expression for S. We see that 



^= coefficient of x"'~^ in the expansion of (1 +£c)"~^ 



m 



Hence S is equal to | m — 2 times the coefficient of cc'" " in the 

 expansion of 



(1 + xy-' + (1 + xy-' +(14- xy-" + .,. 



Now in the game of Pharaon we have n always even ; thus we 

 may suppose the series to be continued down to 1, and then its 

 sum is 



(i+^)"-i ^, . . (1+^r-i 



(1 -\-'xy^l ^^'""^ '' 2x + x' ' 

 Thus we require the coefficient of x"'"^ in the expansion of 



(1 + xy - 1 



2-^x ' 



16— 2 



