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A202678
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A185957; by antidiagonals.
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3
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1, -1, 1, -11, 1, 1, -30, 57, -1, 1, -50, 395, -203, 1, 1, -70, 1133, -3221, 574, -1, 1, -90, 2271, -15840, 19011, -1386, 1, 1, -110, 3809, -45980, 156892, -88729, 2982, -1, 1, -130, 5747, -101640, 660617, -1195097, 346295, -5874
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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 are 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 A185957 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 {{1,3,6},{3,10,21},{6,21,46}}, with p(3)=1-30x+57x^2-x^3 and zero-set {0.035..., 0.495..., 56.469...}.
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Top of the array:
1...-1
1...-11...1
1...-30...57....-1
1...-50...395...-203...1
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MATHEMATICA
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f[k_] := k (k + 1)/2
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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