|
|
A204011
|
|
Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{3i+j-3,i+3j-3} (A204008).
|
|
3
|
|
|
1, -1, -11, -6, 1, 40, 70, 15, -1, -116, -328, -240, -28, 1, 304, 1176, 1456, 610, 45, -1, -752, -3680, -6408, -4704, -1295, -66, 1, 1792, 10592, 23760, 25080, 12432, 2436, 91, -1, -4160, -28800, -79040
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
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
-11....-6.....1
40.....70....15....-1
-116...-328..-240....1
|
|
MATHEMATICA
|
f[i_, j_] := Max[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}]] (* A204008 *)
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
|
|
|
|