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A203956
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203955.
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3
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1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -27, 165, -35, 1, 1, -123, 1255, -511, 54, -1, 1, -300, 9266, -6003, 1197, -82, 1, 1, -558, 77523, -71564, 20779, -2463, 111, -1, 1, -2841, 688624, -817771, 315489, -54393, 4386, -144, 1, 1, -9093
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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 positive, 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...-6....1
1...-12....20....-1
1...-27....165...-35....1
1...-123...1255..-511...54...-1
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MATHEMATICA
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t = {1, 2, 3}; t1 = Flatten[{t, t, t, t, t, t, t, t, t}];
f[k_] := t1[[k]];
U[n_] :=
NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
p[n_] := CharacteristicPolynomial[L[n].U[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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