A202678 - OEIS

%I #10 Oct 02 2017 09:57:58

%S 1,-1,1,-11,1,1,-30,57,-1,1,-50,395,-203,1,1,-70,1133,-3221,574,-1,1,

%T -90,2271,-15840,19011,-1386,1,1,-110,3809,-45980,156892,-88729,2982,

%U -1,1,-130,5747,-101640,660617,-1195097,346295,-5874

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A185957; by antidiagonals.

%C Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they are interlace the zeros of p(n+1).

%H S.-G. Hwang, <a href="http://matrix.skku.ac.kr/Series-E/Monthly-E.pdf">Cauchy's interlace theorem for eigenvalues of Hermitian matrices</a>, American Mathematical Monthly 111 (2004) 157-159.

%H A. Mercer and P. Mercer, <a href="http://dx.doi.org/10.1155/S016117120000257X">Cauchy's interlace theorem and lower bounds for the spectral radius</a>, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

%e The 1st principal submatrix (ps) of A185957 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.

%e ...

%e The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.

%e ...

%e The 3rd ps is {{1,3,6},{3,10,21},{6,21,46}}, with p(3)=1-30x+57x^2-x^3 and zero-set {0.035..., 0.495..., 56.469...}.

%e ...

%e Top of the array:

%e 1...-1

%e 1...-11...1

%e 1...-30...57....-1

%e 1...-50...395...-203...1

%t f[k_] := k (k + 1)/2

%t U[n_] :=  NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];

%t L[n_] := Transpose[U[n]];

%t F[n_] := CharacteristicPolynomial[L[n].U[n], x];

%t c[n_] := CoefficientList[F[n], x]

%t TableForm[Flatten[Table[F[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%]

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A185957, A202605.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Dec 22 2011