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A204122
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(2^(i-1), 2^(j-1)) (A144464).
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2
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1, -1, 1, -3, 1, 2, -8, 7, -1, 8, -36, 43, -15, 1, 64, -304, 414, -198, 31, -1, 1024, -4992, 7224, -3960, 849, -63, 1, 32768, -161792, 241088, -140864, 34674, -3516, 127, -1, 2097152, -10420224, 15752192, -9492480, 2493640, -290412
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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 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;
1, -3, 1;
2, -8, 7, -1;
8, -36, 43, -15, 1;
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MATHEMATICA
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f[i_, j_] := GCD[2^(i - 1), 2^(j - 1)];
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}]] (* A144464 *)
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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