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 A202870 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202869; by antidiagonals. 3
 1, -1, 1, -11, 1, 1, -46, 37, -1, 1, -162, 299, -99, 1, 1, -567, 1675, -1324, 225, -1, 1, -1872, 8316, -11315, 5292, -432, 1, 1, -5881, 40254, -79457, 60782, -16458, 760, -1, 1, -17990, 182413, -490520, 543130, -260498, 45424, -1232 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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). LINKS Table of n, a(n) for n=1..43. S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159. A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566. EXAMPLE The 1st principal submatrix (ps) of A202869 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}. ... The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}. ... The 3rd ps is {{1,3,4},{3,10,15},{4,15,26}}, with p(3)=1-46x+37x^2-x^3 and zero-set {0.022..., 1.265..., 35.712...}. ... Top of the array: 1...-1 1...-11....1 1...-46....37....-1 1...-162...299...-99...1 MATHEMATICA f[k_] := Floor[k*GoldenRatio]; U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]]; L[n_] := Transpose[U[n]]; F[n_] := CharacteristicPolynomial[L[n].U[n], x]; c[n_] := CoefficientList[F[n], x] TableForm[Flatten[Table[F[n], {n, 1, 10}]]] Table[c[n], {n, 1, 12}] Flatten[%] (* A202870 as a sequence *) TableForm[Table[c[n], {n, 1, 10}]] (* A202870 as a matrix *) CROSSREFS Cf. A202869, A202605. Sequence in context: A202971 A202675 A176198 * A202872 A144440 A157209 Adjacent sequences: A202867 A202868 A202869 * A202871 A202872 A202873 KEYWORD tabl,sign AUTHOR Clark Kimberling, Dec 26 2011 STATUS approved

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Last modified September 17 21:09 EDT 2024. Contains 375990 sequences. (Running on oeis4.)