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).

-1, 1, 1, -16, 12, -1, 381, -3312, 2160, 1, -10496, 1603680, -10137600, 6048000, -1, 307505, -1022881200, 92708406000, -476703360000, 266716800000, 1, -9316560, 750409713900, -1242627237734400, 78981336366912000, -349935855575040000, 186313420339200000, -1

Offset: 1

Author

Benoit Cloitre, Nov 27 2002

Keywords

• sign
• tabl

Comments

Montgomery made a conjecture related to the largest eigenvalue of the Hilbert matrix (cf. Matthews link)

Examples

			Triangle begins:
  -1, 1;
  1, -16, 12;
  -1, 381, -3312, -2160;
  ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..902
• Keith Matthews, Hilbert inequality.

Crossrefs

Cf. A005249.

Programs

Maple

f:= proc(n) uses LinearAlgebra; local P,M;
  M:= HilbertMatrix(n);
  P:= CharacteristicPolynomial(M,t)/Determinant(M);
  seq(coeff(P,t,i),i=0..n)
end proc:
seq(f(n),n=1..10); # Robert Israel, May 07 2018

Mathematica

row[n_] := Module[{P, M, x}, M = HilbertMatrix[n]; P = CharacteristicPolynomial[M, x]/Det[M]; (-1)^n CoefficientList[P, x]];
Array[row, 10] // Flatten (* Jean-François Alcover, Jun 22 2020 *)

PARI

vector(n+1,i,(polcoeff(charpoly(mathilbert(n))/matdet(mathilbert(n)),i-1))) \\ for the "n-th row"

Formula

T(n,0)=(-1)^n, T(n,n) = A005249(n). - Robert Israel, May 07 2018