302 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
*1 = « 0 = 1 , 
= «0 + Cl l = 3 > 
.So = a 0 4- a x + a. 2 = 9, 
i.e., the successive values of s n + 2 are the Hamiltonian Numbers 3, 5, 11 , 47 . . . 
Now F„(x) = a n x n , so that the coefficient of x n in (1 — x)*» F„(x) is the same 
as the coefficient of x" in F„(x), viz., a„. Consequently, equating coefficients of x” on 
each side of the equation just obtained, we find 
a, 
- i + (*, + i) = 
(£»-i + 2) (s« _ 1 + 1) s, t - i 
1.2.3 
Remembering that 
+ . 
■ / \ n +1 (^i + 2) (g T + 1 ) . . . (s T + 2 ri) 
^ ' ’ 1 . 2.3 ... (n + 1 ) 
(X'/i 4" A 1 — S/j + i) 
if we call the Hamiltonian Number s H -j- 2 , H„, the above relation may be written 
thus : 
xx 0 (H„ — 1) H m _ t (H«_ 1 — 1) (H„_ 1 — 2) 
H " + 1 “ 2= hi - " lThNr 
H„_ 8 (H„_ 2 — 1) (H a _ 2 — 2) (H„_ 2 — 3) 
~ ~ 1 . 2 . 3.4 
, / _ wi H t (H I - 1) (H, - 2) . . . (H, - ») 
^ ' ’ 1.2.3 . . . (n + 1) 
To obtain Professor Sylvester’s modification of this formula given in the preceding 
portion of this memoir, we multiply the equation from which it was obtained by 
1 — x before proceeding to equate coefficients. Thus we have to equate coefficients 
of x n on both sides of 
(1 — x) Sn +1 F« (x) — 1 -p x~ ] (l — x) 3 — x" -1 (1 — x) s “ +3 
= x" -3 (l — x )*« +3 fi- x w-3 (l — x ) ,s “' 1 + 3 + x" _4 (l — x y n -2+z 4- ... 4 " 3 ; _1 — x>" + °. 
Or, writing 
•V + 3 = E,„ 
we equate coefficients on both sides of 
(1 — x) E "- 3 F„ (x) — 1 -j- x*" 1 (1 — x) 3 — x w_1 (l — x) E “"“ 1 
= x"~~ (L — x) E » 4 x n "" 3 (1 — x)®"- 1 + x”“ 4 (1 — x) E "- a 4 . . . x” 1 (1 — x) E| . 
