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A204023
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-1, 2j-1) (A204022).
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3
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1, -1, -6, -4, 1, 20, 36, 9, -1, -56, -160, -120, -16, 1, 144, 560, 700, 300, 25, -1, -352, -1728, -3024, -2240, -630, -36, 1, 832, 4928, 11088, 11760, 5880, 1176, 49, -1, -1920, -13312, -36608, -50688, -36960
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OFFSET
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1,3
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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 real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.
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REFERENCES
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(For references regarding interlacing roots, see A202605.)
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LINKS
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EXAMPLE
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Top of the array:
1....-1
-6....-4.....1
20....36....9.....-1
-56...-160..-120...-16....1
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MATHEMATICA
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f[i_, j_] := Max[2 i - 1, 2 j - 1];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A204022 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
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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