A202677 - OEIS

%I #12 Oct 02 2017 09:57:51

%S 1,-1,1,-18,1,1,-116,84,-1,1,-538,1347,-250,1,1,-2256,11566,-8216,585,

%T -1,1,-9158,75453,-118722,35086,-1176,1,1,-36796,426288,-1152432,

%U 801084,-118656,2128,-1,1,-147378,2214919,-8910538,11175711,-4079622,339762,-3564,1,1,-589736,10915650

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202676 based on (1,4,7,10,13,...); 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 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 A202676 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.

%e ...

%e The 2nd ps is {{1,4},{4,17}}, with p(2)=1-18x+x^2 and zero-set {0.055..., 17.944...}.

%e ...

%e The 3rd ps is {{1,4,7},{4,17,32},{7,32,66}}, with p(3)=1-116x+84x^2-x^3 and zero-set {0.008..., 1.395..., 82.595...}.

%e ...

%e Top of the array:

%e 1...-1

%e 1...-18....1

%e 1...-116...84.....-1

%e 1...-538...1347...-250...1

%t f[k_] := 3 k - 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. A202676, A202677, A202605.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Dec 22 2011