|
|
A204013
|
|
Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{3i+j-3,i+3j-3} (A204012).
|
|
3
|
|
|
1, -1, 1, -6, 1, 0, -10, 15, -1, -4, -8, 40, -28, 1, -16, 24, 56, -110, 45, -1, -48, 160, -72, -224, 245, -66, 1, -128, 608, -880, 120, 672, -476, 91, -1, -320, 1920, -4160, 3520, 0, -1680, 840, -120, 1, -768, 5504, -15360, 20384
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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 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:
1....-1
1....-6....1
0....-10...15....-1
-4....-8....40....-28....1
|
|
MATHEMATICA
|
f[i_, j_] := Min[3 i + j - 3, 3 j + i - 3];
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}]] (* A204012 *)
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}]]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|