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%I #16 Jun 22 2020 07:23:27
%S -1,1,1,-16,12,-1,381,-3312,2160,1,-10496,1603680,-10137600,6048000,
%T -1,307505,-1022881200,92708406000,-476703360000,266716800000,1,
%U -9316560,750409713900,-1242627237734400,78981336366912000,-349935855575040000,186313420339200000,-1
%N Array of coefficients of 1/det(M_n)*P(M_n) where P(M_n) is the characteristic polynomial of M_n, the n-th n X n Hilbert matrix M_n(i,j)=1/(i+j-1).
%C Montgomery made a conjecture related to the largest eigenvalue of the Hilbert matrix (cf. link)
%H Robert Israel, <a href="/A076823/b076823.txt">Table of n, a(n) for n = 1..902</a>
%H Keith Matthews, <a href="http://www.numbertheory.org/pdfs/hilbert.pdf">Hilbert inequality</a>.
%F T(n,0)=(-1)^n, T(n,n) = A005249(n). - _Robert Israel_, May 07 2018
%p f:= proc(n) uses LinearAlgebra; local P,M;
%p M:= HilbertMatrix(n);
%p P:= CharacteristicPolynomial(M,t)/Determinant(M);
%p seq(coeff(P,t,i),i=0..n)
%p end proc:
%p seq(f(n),n=1..10); # _Robert Israel_, May 07 2018
%t row[n_] := Module[{P, M, x}, M = HilbertMatrix[n]; P = CharacteristicPolynomial[M, x]/Det[M]; (-1)^n CoefficientList[P, x]];
%t Array[row, 10] // Flatten (* _Jean-François Alcover_, Jun 22 2020 *)
%o (PARI) vector(n+1,i,(polcoeff(charpoly(mathilbert(n))/matdet(mathilbert(n)),i-1) \\ for the "n-th row"
%Y Cf. A005249.
%K sign,tabl
%O 1,4
%A _Benoit Cloitre_, Nov 27 2002
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