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A204155
Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-j, 2j-i), as in A204154.
3
1, -1, -7, -3, 1, 33, 39, 6, -1, -135, -255, -125, -10, 1, 513, 1323, 1092, 305, 15, -1, -1863, -6075, -7047, -3444, -630, -21, 1, 6561, 25839, 38610, 27135, 8946, 1162, 28, -1, -22599, -104247, -190593, -175230
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 and A204016 for guides to related sequences.
REFERENCES
(For references regarding interlacing roots, see A202605.)
LINKS
Robert Israel, Table of n, a(n) for n = 1..10010 (rows 1 to 140, flattened)
EXAMPLE
Top of the array:
1, -1;
-7, -3, 1;
33, 39, 6, -1;
-135, -255, -125, -10, 1;
MAPLE
f:= proc(n) local P, lambda, i;
P:= (-1)^n*LinearAlgebra:-CharacteristicPolynomial(Matrix(n, n, (i, j) -> max(2*i-j, 2*j-i)), lambda);
seq(coeff(P, lambda, i), i=0..n);
end proc:
map(f, [$1..20]); # Robert Israel, Dec 03 2017
MATHEMATICA
f[i_, j_] := Max[2 i - j, 2 j - i];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A204154 *)
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[%] (* A204155 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
tabl,sign,look
AUTHOR
Clark Kimberling, Jan 12 2012
STATUS
approved