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The determinant of an n X n matrix derived from the matrix X(s,k) = s^2 - 2*s + k.
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%I #11 Aug 10 2022 09:44:48

%S 0,-3,42,-532,7280,-110816,1878912,-35290368,729151488,-16458706944,

%T 403306168320,-10667511152640,303026291343360,-9203027198607360,

%U 297626965251194880,-10212876839131545600,370647479637717811200,-14185745639287868620800,571060601049504861388800

%N The determinant of an n X n matrix derived from the matrix X(s,k) = s^2 - 2*s + k.

%C Start from an auxiliary infinite matrix X with entries X(s,k) = s^2 - 2*s + k, with indices s, k >= 1.

%C Build a matrix T by preserving the subdiagonal triangular part of X and filling the superdiagonal triangular part of T by reading X with a 90-degree hook at the diagonal: T(m,k) = X(m,k) if m >= k and T(m,k) = X(k,2k-m) if m < k. Imagine reading X along columns along decreasing row number and continuing along increasing column number after reaching the diagonal. Then a(n) = determinant(T).

%F a(n) = (-2)^(n-2)*(n-1)!*((Sum_{k=1..n-1} 1/k) - 2*n*(n-1)).

%p with(linalg) : X := proc(s,k) s^2-2*s+k ; end proc:

%p A173936 := proc(n) T := matrix(n,n) ; for m from 1 to n do for k from 1 to n do if m >= k then T[m,k] := X(m,k) ; else T[m,k] := X(k,2*k-m) ; end if; end do ; end do ; det(T) ; end proc:

%p seq(A173936(n),n=1..20) ; # _R. J. Mathar_, Mar 05 2010

%K sign

%O 1,2

%A Anonymous, Mar 03 2010

%E Extended and description simplified by _R. J. Mathar_, Mar 05 2010