MONTMORT. 89 



156. We are thus naturally led to consider the summation of 

 certain series. 



Let (71, r) = -^ ^ 



so that (j) {n, r) is the n^^ number of the (r + 1)"' order of figurate 

 numbers. 



Let 8<f) (n, r) stand for <j) {n, r) + <^ (w — 2, ?•) + </> (^ — 4, r) + . . . , 

 so that S(j> (n, r) is the sum of the alternate terms of the series of 

 figurate numbers of the (r + 1)**" order, beginning with the w"' and 

 going backwards. It is required to find an expression for /S'</) {n, r). 

 It is known that 



(n, r) + (l)(n-l,r) +(f) {n - 2, r) + </> (71 - 3, r) +... = (/> (ji, r + 1) ; 



and by taking the terms in pairs it is easy to see that 



<j) (n, 7') — (j) (n — l,r) -{-(f) [n — 2, 7^) —(f>{n — 3, ?•) + ... = S(j) {n, r — 1) ; 



therefore, by addition, 



S(l> {n, r) = - (/) {71, r-\-l)+^S(i> (w, r- 1). 



Hence, continuing the process, we shall have 



1 11 



^^ (w, ?•) = 2 ^ ^^' ** + ^) + 3 ^ (^^' ^') + ^ <^ {^h r - 1) + ... 



and we must consider 8<\> (n, ^—-n, if 71 be even, and = - (n+1), 

 if n be odd. 



We may also obtain another expression for 8<^ {n, r). For 

 change w into n + 1 in the two fundamental relations, and subtract, 

 instead of adding as before ; thus 



>^(/) (71, r) = i <^ (n + 1 , r + 1) - ^ ^0 ( ; . + 1 , r - 1 ) . 

 Hence, continuing the process, we shall have 

 ^(/)(7i,r)=-(/>(7i + l, r + 1)- ^ </,(« + 2,7')+^ <3«>(n + 3, r-1) 



{- ly {- \Y 



- -^ ^(u + r, 2) + 4^ Sct>(n + r, 0). 



