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A204176
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is even, and 0 otherwise) as in A204175.
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2
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0, -1, -1, -1, 1, 0, 1, 1, -1, 1, 1, -3, -2, 1, 0, -1, -1, 3, 2, -1, -1, -1, 5, 4, -6, -3, 1, 0, 1, 1, -5, -4, 6, 3, -1, 1, 1, -7, -6, 15, 10, -10, -4, 1, 0, -1, -1, 7, 6, -15, -10, 10, 4, -1, -1, -1, 9, 8, -28, -21, 35, 20, -15, -5, 1, 0, 1, 1, -9, -8, 28, 21
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OFFSET
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1,12
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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:
0....-1
-1....-1.....1
0.....1.....1....-1
1.....1....-3....-2...1
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MATHEMATICA
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f[i_, j_] := If[Mod[Max[i, j], 2] == 0, 1, 0]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A204175 *)
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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tabf,sign
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AUTHOR
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STATUS
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approved
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