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A176113 Determinant of n X n matrix whose (i,j)-th element is (i+j)^(n-1).

%I A176113 #17 Jun 02 2025 02:49:26
%S A176113 1,-1,-8,1296,7962624,-2985984000000,-100306130042880000000,
%T A176113 416336312719673760153600000000,
%U A176113 281633758444745849464726940024832000000000
%N A176113 Determinant of n X n matrix whose (i,j)-th element is (i+j)^(n-1).
%C A176113 a(n) = A057077(n)*A091868(n-1), signed variant of A091868.
%C A176113 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 :
%C A176113 x = 1! * 2!*...(n-1)! ;
%C A176113 y = (u(1) - u(2))*((u(1) - u(3))*....*(u(n-1) - u(n)) ;
%C A176113 z = (v(n) - v(n-1))*((v(n) - v(n-2))*....*(v(2) - v(1)).
%C A176113 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.
%D A176113 J. M. Monier, Algebre & geometrie, Dunod (1996), p.216.
%F A176113 a(n) = (-1)^floor(n/2)* ((n-1)!)^n.
%e A176113 a(3) = determinant(M_3) = -8 where M_3 is the matrix
%e A176113 [4  9   16]
%e A176113 [9  16  25]
%e A176113 [16 25  36]
%t A176113 a[n_] = ((-1)^((n - Mod[n, 2])/2))*((n - 1)!)^n; Table[a[n], {n, 9}] (* from Jean-François Alcover, Aug 31 2011 *)
%Y A176113 Cf. A091868.
%K A176113 sign
%O A176113 1,3
%A A176113 _Michel Lagneau_, Apr 08 2010