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A203991 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {(i+j)*min(i,j)} (A203990). 3
2, -1, 7, -10, 1, 38, -71, 28, -1, 281, -610, 357, -60, 1, 2634, -6329, 4620, -1253, 110, -1, 29919, -77530, 65613, -23348, 3514, -182, 1, 399342, -1098271, 1036044, -442349, 90800, -8442, 280, -1, 6125265 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
REFERENCES
(For references regarding interlacing roots, see A202605.)
LINKS
EXAMPLE
Top of the array:
2.... -1
7.... -10... 1
38... -71... 28... -1
281.. -610.. 357.. -60... 1
MATHEMATICA
f[i_, j_] := (i + j) Min[i, j];
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}]] (* A203990 *)
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}]
Flatten[%] (* A203991 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
Sequence in context: A032210 A032135 A032039 * A075118 A100245 A275320
KEYWORD
tabl,sign
AUTHOR
Clark Kimberling, Jan 09 2012
STATUS
approved

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Last modified May 19 13:25 EDT 2024. Contains 372694 sequences. (Running on oeis4.)