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A202872
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202871; by antidiagonals.
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2
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1, -1, 1, -11, 1, 1, -46, 37, -1, 1, -181, 298, -112, 1, 1, -716, 1784, -1368, 308, -1, 1, -2851, 9495, -11119, 5286, -828, 1, 1, -11386, 47431, -74940, 55235, -18546, 2189, -1, 1, -45521, 227592, -453206, 455080, -239360, 61185, -5759
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OFFSET
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1,4
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COMMENTS
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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).
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LINKS
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EXAMPLE
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The 1st principal submatrix (ps) of A202871 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
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The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
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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...}.
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Top of the array:
1...-1
1...-11....1
1...-46....37....-1
1...-181...298...-112...1
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MATHEMATICA
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f[k_] := LucasL[k];
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[%]
TableForm[Table[c[n], {n, 1, 10}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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