A076823 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).

-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

1,4

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..902

Keith Matthews, Hilbert inequality.

Formula

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

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 *)

Prog

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

Crossrefs

Cf. A005249.

Sequence in context: A298450 A028695 A008665 * A070551 A028493 A291426

Adjacent sequences: A076820 A076821 A076822 * A076824 A076825 A076826

Keyword

sign,tabl

Author

Benoit Cloitre, Nov 27 2002

Status

approved