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A203946
Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203945.
3
1, -1, 1, -2, 1, 1, -3, 3, -1, 1, -5, 8, -5, 1, 1, -7, 17, -17, 7, -1, 1, -9, 30, -45, 30, -9, 1, 1, -11, 47, -98, 103, -52, 12, -1, 1, -13, 68, -183, 269, -212, 83, -15, 1, 1, -15, 93, -308, 588, -651, 399, -123, 18, -1, 1, -17, 122, -481, 1136
OFFSET
1,4
COMMENTS
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.
REFERENCES
(For references regarding interlacing roots, see A202605.)
EXAMPLE
Top of the array:
1...-1
1...-2....1
1...-3....3....-1
1...-5....8....-5....1
1...-7....17...-17...7...-1
MATHEMATICA
t = {1, 0, 0}; t1 = Flatten[{t, 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}]
Flatten[%] (* A203946 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
Sequence in context: A089980 A181031 A214987 * A128545 A194672 A034364
KEYWORD
tabf,sign
AUTHOR
Clark Kimberling, Jan 08 2012
STATUS
approved