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A204019
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{1+j mod i, 1+i mod j} (A204018).
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3
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1, -1, -3, -2, 1, 8, 14, 3, -1, -21, -64, -40, -4, 1, 40, 266, 280, 90, 5, -1, 125, -930, -1671, -896, -175, -6, 1, -2940, 854, 8600, 7228, 2352, 308, 7, -1, 35035, 37744, -27334, -50164, -24594, -5376, -504, -8, 1, -372400
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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). The least zero of p(n) is -n.
For n>1, the least zero of p(n) is exactly 1-n; the greatest, for p(1) to p(5) is represented by (1,3,5.701...,9.158...13.392...).
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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
-3....-2......1
8.....14.....3....-1
-21...-64....-40...-4...1
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MATHEMATICA
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f[i_, j_] := 1 + Max[Mod[i, j], Mod[j, i]];
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}]] (* A204018 *)
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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