A135451 Triangular function from the characteristic polynomials of the inverse Hilbert matrices.

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

Offset: 0

Author

Roger L. Bagula, Dec 14 2007

Keywords

• tabl
• sign

Comments

Triangle read by rows: for 0 <= k <= n, T(n,k) is the coefficient of lambda^k in det(H^(-1) - lambda I) where H is the n x n Hilbert matrix.

Row sums are: 1, 0, -3, -772, -2496415, -118300727696, -85882975706265059, -972835586209103886374316, -173520203650301344466515679407359, -489847775570499454780372858733881836257416, -21954569246037949585920541114453120558720536422853379

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 0..902 (rows 0 to 41, flattened)
• Eric Weisstein's World of Mathematics, Hilbert matrix

Crossrefs

Cf. A005249.

Programs

Maple

f:= proc(n) uses LinearAlgebra;
local lambda, P,j;
P:= CharacteristicPolynomial(HilbertMatrix(n),lambda)/Determinant(HilbertMatrix(n));
seq(coeff(P,lambda,n-j),j=0..n);
end proc:
seq(f(n),n=0..10); # Robert Israel, Oct 05 2016

Mathematica

<< LinearAlgebra`MatrixManipulation`; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[Inverse[HilbertMatrix[n]], x], x], {n, 1, 10}]]; Flatten[a]

Formula

t(n,m)=CoefficientList[CharacteristicPolynomial[Inverse[HilbertMatrix[n]], x], x]

Extensions

Edited by Robert Israel, Oct 05 2016