The theory of determinants in the historical order of development, by Sir Thomas Muir.

282 HISTORY OF THE THEORY OF DETERMINANTS
and thence the values of u2, u4, u6, u.... as
1, 6, 120, 5250,.....
In the former case the values appear of course as sums of multiples
of u2, u,....; for example, for the 7th order the value is
1 + C7, 1 + C7,3-6+ C7,1.120;
and the first seven values are
1, 2, 4, 13, 41, 226, 1072,....
SYLVESTER, J. J. (1879).
[Sur la valeur moyenne des coefficients dans le developpement
d'un determinant gauche ou sym6trique.... Comptes Rendus.... Acad. des Sci. (Paris), lxxxix. pp. 24-26, 497-498; or
Collected Math. Papers, iii. pp. 253-255, 257.]
In this is contained an improved form of one of the results just
chronicled, namely, that the number of distinct terms in a zeroaxial skew determinant of order 2n is
1 3-5.... (2n —1) vn,
where v. is determined from the equations
vn = (2n-1)v_-1 - (n-1 ) Vn_2, VO =1, v = 1,
or from the equation (corrected)
vn= {l+ n + 15C, +1.5. 9C, +....
A similar statement is made in regard to ' dterminants doublement
gauches,' that is to say, determinants which are skew with respect
to both diagonals and have both diagonals full of zeros. In such
a determinant of the order 4n Sylvester says the number of distinct
terms is
2 4 6.... (4n-2) - w,
where wn is determined from the equations
wn = (4n-3)w,, — 2nw-_2, Wo = 1, w1 = 1,
or from the equation
w,= {l+n+1- 9Cn,2+1-9-7C17 3+...}2n