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A204167
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of ceiling((i+j)/2), as in A204166.
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3
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1, -1, -2, -3, 1, 1, 6, 6, -1, 0, -4, -16, -10, 1, 0, 0, 15, 32, 15, -1, 0, 0, 0, -36, -60, -21, 1, 0, 0, 0, 0, 84, 100, 28, -1, 0, 0, 0, 0, 0, -160, -160, -36, 1, 0, 0, 0, 0, 0, 0, 300, 240, 45, -1, 0, 0, 0, 0, 0, 0, 0, -500, -350
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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
-2, -3, 1
1, 6, 6, -1
0, -4, -16, -10, 1
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MATHEMATICA
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f[i_, j_] := Ceiling[(i + j)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A204166 *)
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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EXTENSIONS
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STATUS
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approved
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