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A203995
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i-j+1,j-i+1} (A203994).
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4
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1, -1, 1, -2, 1, 0, -2, 3, -1, -4, 8, 0, -4, 1, -16, 56, -56, 10, 5, -1, -48, 224, -360, 224, -35, -6, 1, -128, 736, -1584, 1560, -672, 84, 7, -1, -320, 2176, -5824, 7744, -5280, 1680, -168, -8, 1, -768, 6016, -19200, 32032, -29744
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OFFSET
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1,4
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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
1...-2....1
0...-2....3...-1
-4....8....0...-4....1
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MATHEMATICA
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f[i_, j_] := Min[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}]] (* A203994 *)
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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