Note on Recurrents Resolvable into a Sequence of Odd Integers. 31 
which enables us to remove from the last row the factor ^n + 2m — 5 and 
leave the row 
(2n -i- m — 4)„-.2 , (2n -f m — 6)„_3 (m -f 2)^ , 1, 0 
thus giving us 
{2n - 2) n,,{m) = - {2n - 2) • (2/^ + 2m - 5) E,_ ,(m + 2) 
from which we have only to remove the common factor. 
By applying (YIII) to itself and repeating the operation we obtain 
E„(w) = (-l)2(2^i + 2m-5) (27i-i-2m-3) E„_2 (m + 4) 
= (2?^ + 2m-5) (4^n^2m-9) . Ri(m + 2w-2) 
and is 1 for all arguments. 
(11) The complicated operation w^hich brings about the removal of the 
factor 2n-\-2m—5 in the preceding paragraph is based on a property of the 
numbers 1, 3, 10, 35, . . . (2n — l)u—i, which, on the other hand, is simple 
and orderly, namely, 
( 2^ + 2 + g),,+i-(2jj + 3),,+i 
q - 1 
= {2p+l)yq,fl^{2p-l),^r(q + 2\+l(2p-S),^,-{q + 4),-i-. . . 
+ -^TT (^)o^^ + ^^^P .... (IX) 
where the possibility of division on the left-hand side is attested by the 
vanishing of the numerator when q is put equal to 1. It may be viewed as 
a generalisation of the theorem quoted in § 8 : for on putting q = — 1 the 
left-hand member here becomes 
(■2y + 3)„.n - {2p + l)„^ , . 3p + 4 (0,.,.) 
2 2(J+2y^^^'^^>''' > 
1 
after which we have only got to add 2'^^-^^ '^)p order to complete the 
transformation. 
(12) R' may be generalised in its last row exactly as R„ has been, the 
resulting determinant being denoted ])y R;',(«i), and there being a series of 
companion theorems. For example, as analogues to ;^VI) and (VII) we 
have 
R;(m) ^ RL(m - 1) -f- 2(n - l)K>-i(>u + 1) . (X) 
R;(m) - (2n + 2m - 3) (2n + 2m - 5) . . . (4??. + 2m - 7) . . (XI). 
(13) With the closely resembling values of R„('i)i) and 'R>',i(m) before us 
it is easy to suggest relations the proofs of which would form interesting 
exercises in the transformation of determinants. Thus, since from § 9 we 
have 
