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A076823 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).

%I A076823 #26 Apr 17 2025 16:59:20
%S A076823 -1,1,1,-16,12,-1,381,-3312,2160,1,-10496,1603680,-10137600,6048000,
%T A076823 -1,307505,-1022881200,92708406000,-476703360000,266716800000,1,
%U A076823 -9316560,750409713900,-1242627237734400,78981336366912000,-349935855575040000,186313420339200000,-1
%N A076823 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 A076823 Montgomery made a conjecture related to the largest eigenvalue of the Hilbert matrix (cf. Matthews link)
%H A076823 Robert Israel, <a href="/A076823/b076823.txt">Table of n, a(n) for n = 1..902</a>
%H A076823 Keith Matthews, <a href="http://www.numbertheory.org/PDFS/hilbert.pdf">Hilbert inequality</a>.
%F A076823 T(n,0)=(-1)^n, T(n,n) = A005249(n). - _Robert Israel_, May 07 2018
%e A076823 Triangle begins:
%e A076823   -1, 1;
%e A076823   1, -16, 12;
%e A076823   -1, 381, -3312, -2160;
%e A076823   ...
%p A076823 f:= proc(n) uses LinearAlgebra; local P,M;
%p A076823   M:= HilbertMatrix(n);
%p A076823   P:= CharacteristicPolynomial(M,t)/Determinant(M);
%p A076823   seq(coeff(P,t,i),i=0..n)
%p A076823 end proc:
%p A076823 seq(f(n),n=1..10); # _Robert Israel_, May 07 2018
%t A076823 row[n_] := Module[{P, M, x}, M = HilbertMatrix[n]; P = CharacteristicPolynomial[M, x]/Det[M]; (-1)^n CoefficientList[P, x]];
%t A076823 Array[row, 10] // Flatten (* _Jean-François Alcover_, Jun 22 2020 *)
%o A076823 (PARI) vector(n+1,i,(polcoeff(charpoly(mathilbert(n))/matdet(mathilbert(n)),i-1))) \\ for the "n-th row"
%Y A076823 Cf. A005249.
%K A076823 sign,tabl
%O A076823 1,4
%A A076823 _Benoit Cloitre_, Nov 27 2002