Note on Recurrents Resolvable into a Sequence of Odd Integers. 29 
(5) Similarly we may treat R^^ as R„ has been treated in § 3, the 
difference being that the addition of the sum of multiples of rows is made 
to —3 times the n^^ row, and that the factor removed is —(4<n—l). 
Calling the new determinant U^^, which, be it noted, has the same last row 
as JJn , we have 
U|^= 3 (2^ + 3) (2/^ + 5) (4>n-S) . . (Ill) 
(6) Lastly, we may treat as R„ has been treated in § 4, the difference 
being that the addition of the sum of multiples of rows is made to - 5 times 
the n^^ row, and that the factor removed is -(2?i + 3) : so that if the 
resulting determinant be denoted by V^^ we have 
5 (2n + 5) (2n +7) .... (4.. - 1) ; . . (lY) 
for example, 
1 -2 . 
3 1-4. 
10 3 1-6. 
35 10 3 1 -8 
42 14 5 2 1 
(7) Comparing the equalities obtained in §§4, 5, we reach the otherwise 
curious result 
• • • ■ (V) 
V 
515-17-19. 
u; = (-i)»-i v„; 
for example. 
1 
-2 
1 
2 
3 
1 
-4 
3 
1 
4 
10 
3 
1 
-6 
10 
3 
1 
6 
. 
35 
10 
3 
1 
-8 
35 
10 
3 
1 
8 
14 
5 
2 
1 
1 
42 
14 
5 
2 
1 
(8) The four equalities upon which depends the finding of the new 
n^h. rows in §§ 3, 4, 5, 6 are properties of the series 
1, 3, 10, 35, 126, 
referred to in § 1. They are all, however, reducible to one fundamental 
equality, namely, 
1 1 1 
(2m + 1)„, + 2" (2^'^ 'lu + 3(2m-3)„_2-3i + . . . +^^— ^(2m + l); 
or, if we call the members of the series t^ , , . 
11 1 Yf / 
\i, 2' 3' 
m 
' '^i ? 1X1' ^1 
, t 
111 + 1 
3 
It is not essentially different from Reich's principal equality (p. 181) 
which he proves gradationally, that is to say, by proceeding from one value 
of 771 to the next higher. 
