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A203992
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (A143182 in square format).
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2
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1, -1, -3, -2, 1, 8, 14, 3, -1, -20, -56, -40, -4, 1, 48, 184, 224, 90, 5, -1, -112, -544, -936, -672, -175, -6, 1, 256, 1504, 3344, 3480, 1680, 308, 7, -1, -576, -3968, -10816, -14784, -10560, -3696, -504, -8, 1, 1280, 10112, 32640, 55328
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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 for a guide 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
-3... -1.... 1
8.... 14... 3... -1
-20.. -56.. -40.. -4... 1
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MATHEMATICA
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f[i_, j_] := Max[i - j + 1, j - i + 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, 12}, {i, 1, n}]] (* A143182 in square format *)
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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