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A202678 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A185957; by antidiagonals.

%I A202678 #10 Oct 02 2017 09:57:58
%S A202678 1,-1,1,-11,1,1,-30,57,-1,1,-50,395,-203,1,1,-70,1133,-3221,574,-1,1,
%T A202678 -90,2271,-15840,19011,-1386,1,1,-110,3809,-45980,156892,-88729,2982,
%U A202678 -1,1,-130,5747,-101640,660617,-1195097,346295,-5874
%N A202678 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A185957; by antidiagonals.
%C A202678 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 A202678 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 A202678 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 A202678 The 1st principal submatrix (ps) of A185957 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A202678 ...
%e A202678 The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
%e A202678 ...
%e A202678 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 A202678 ...
%e A202678 Top of the array:
%e A202678 1...-1
%e A202678 1...-11...1
%e A202678 1...-30...57....-1
%e A202678 1...-50...395...-203...1
%t A202678 f[k_] := k (k + 1)/2
%t A202678 U[n_] :=  NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202678 L[n_] := Transpose[U[n]];
%t A202678 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202678 c[n_] := CoefficientList[F[n], x]
%t A202678 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202678 Table[c[n], {n, 1, 12}]
%t A202678 Flatten[%]
%t A202678 TableForm[Table[c[n], {n, 1, 10}]]
%Y A202678 Cf. A185957, A202605.
%K A202678 tabl,sign
%O A202678 1,4
%A A202678 _Clark Kimberling_, Dec 22 2011