|
|
A202870
|
|
Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202869; by antidiagonals.
|
|
3
|
|
|
1, -1, 1, -11, 1, 1, -46, 37, -1, 1, -162, 299, -99, 1, 1, -567, 1675, -1324, 225, -1, 1, -1872, 8316, -11315, 5292, -432, 1, 1, -5881, 40254, -79457, 60782, -16458, 760, -1, 1, -17990, 182413, -490520, 543130, -260498, 45424, -1232
(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 positive, and they interlace the zeros of p(n+1).
|
|
LINKS
|
|
|
EXAMPLE
|
The 1st principal submatrix (ps) of A202869 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
...
The 3rd ps is {{1,3,4},{3,10,15},{4,15,26}}, with p(3)=1-46x+37x^2-x^3 and zero-set {0.022..., 1.265..., 35.712...}.
...
Top of the array:
1...-1
1...-11....1
1...-46....37....-1
1...-162...299...-99...1
|
|
MATHEMATICA
|
f[k_] := Floor[k*GoldenRatio];
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%] (* A202870 as a sequence *)
TableForm[Table[c[n], {n, 1, 10}]] (* A202870 as a matrix *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|