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%I #15 Oct 02 2017 09:36:43
%S 1,-1,1,-6,1,1,-12,20,-1,1,-18,75,-50,1,1,-24,166,-328,105,-1,1,-30,
%T 293,-1050,1134,-196,1,1,-36,456,-2432,5140,-3312,336,-1,1,-42,655,
%U -4690,15471,-20814,8514,-540,1,1,-48,890,-8040,36771,-80584
%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115263 based on (1,2,3,4,...); by antidiagonals.
%C Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix of A115262 (when A115262 is formatted as a square matrix). The zeros of p(n) are positive, and they 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 A115263 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e ...
%e The 2nd ps is {{1,2},{2,5}}, with p(2)=1-6x+x^2 and zero-set {0.171..., 5.828...}.
%e ...
%e The 3rd ps is {{1,2,3},{2,5,8},{3,8,14}}, with p(3)=1-12x+20x^2-x^3 and zero-set {0.099..., 0.516..., 19.383...}.
%e ...
%e Top of the array:
%e 1....-1
%e 1....-6.....1
%e 1...-12....20.....-1
%e 1...-18....75....-50....1
%e 1...-24...166...-328..105..-1
%t U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[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. A115262, A202671 (based on n^2), A202605 (based on Fibonacci numbers)
%K tabl,sign
%O 1,4
%A _Clark Kimberling_, Dec 22 2011
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