OFFSET
1,3
COMMENTS
Concerning the general case det((u(i) + v(j))^(n-1) for 1<=i,j<=n and u(1), ...,u(n), v(1), ..., v(n) integers, the reference give the proof that det((u(i) + v(j))^(n-1) = (1/x) *(((n-1)!)^n)*y*z with the following results :
x = 1! * 2!*...(n-1)! ;
y = (u(1) - u(2))*((u(1) - u(3))*....*(u(n-1) - u(n)) ;
z = (v(n) - v(n-1))*((v(n) - v(n-2))*....*(v(2) - v(1)).
If the (i,j)-th element is (i+j)^(n-1), then u(i) = i and v(j) = n+1-j. Finally, det(n X n) = ((-1)^p)* (n!)^(n+1) with n = 2p + 1 or n=2p.
REFERENCES
J. M. Monier, Algebre & geometrie, Dunod (1996), p.216.
FORMULA
a(n) = (-1)^floor(n/2)* ((n-1)!)^n.
EXAMPLE
a(3) = determinant(M_3) = -8 where M_3 is the matrix
[4 9 16]
[9 16 25]
[16 25 36]
MATHEMATICA
a[n_] = ((-1)^((n - Mod[n, 2])/2))*((n - 1)!)^n; Table[a[n], {n, 9}] (* from Jean-François Alcover, Aug 31 2011 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Michel Lagneau, Apr 08 2010
STATUS
approved