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Array of coefficients of 1/det(M_n)*P(M_n) where P(M_n) is the characteristic polynomial of the n-th n X n Hilbert matrix M_n(i,j)=1/(i+j-1).
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%I #24 Aug 22 2024 02:48:46

%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 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