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

%I A135451 #13 Feb 16 2025 08:33:07
%S A135451 1,1,-1,12,-16,1,2160,-3312,381,-1,6048000,-10137600,1603680,-10496,1,
%T A135451 266716800000,-476703360000,92708406000,-1022881200,307505,-1,
%U A135451 186313420339200000,-349935855575040000,78981336366912000,-1242627237734400,750409713900,-9316560,1
%N A135451 Triangular function from the characteristic polynomials of the inverse Hilbert matrices.
%C A135451 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.
%C A135451 Row sums are: 1, 0, -3, -772, -2496415, -118300727696, -85882975706265059, -972835586209103886374316, -173520203650301344466515679407359, -489847775570499454780372858733881836257416, -21954569246037949585920541114453120558720536422853379
%H A135451 Robert Israel, <a href="/A135451/b135451.txt">Table of n, a(n) for n = 0..902</a>  (rows 0 to 41, flattened)
%H A135451 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertMatrix.html">Hilbert matrix</a>
%F A135451 t(n,m)=CoefficientList[CharacteristicPolynomial[Inverse[HilbertMatrix[n]], x], x]
%e A135451 {1},
%e A135451 {1, -1},
%e A135451 {12, -16, 1},
%e A135451 {2160, -3312, 381, -1},
%e A135451 {6048000, -10137600, 1603680, -10496, 1},
%e A135451 {266716800000, -476703360000, 92708406000, -1022881200, 307505, -1},
%e A135451 {186313420339200000, -349935855575040000, 78981336366912000, -1242627237734400, 750409713900, -9316560, 1}
%p A135451 f:= proc(n) uses LinearAlgebra;
%p A135451 local lambda, P,j;
%p A135451 P:= CharacteristicPolynomial(HilbertMatrix(n),lambda)/Determinant(HilbertMatrix(n));
%p A135451 seq(coeff(P,lambda,n-j),j=0..n);
%p A135451 end proc:
%p A135451 seq(f(n),n=0..10); # _Robert Israel_, Oct 05 2016
%t A135451 << LinearAlgebra`MatrixManipulation`; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[Inverse[HilbertMatrix[n]], x], x], {n, 1, 10}]]; Flatten[a]
%Y A135451 Cf. A005249.
%K A135451 tabl,sign
%O A135451 0,4
%A A135451 _Roger L. Bagula_, Dec 14 2007
%E A135451 Edited by _Robert Israel_, Oct 05 2016