A204172 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is odd, and 0 otherwise) as in A204171.

1, -1, 0, -1, 1, -1, 1, 2, -1, 0, 1, -1, -2, 1, 1, -1, -4, 3, 3, -1, 0, -1, 1, 4, -3, -3, 1, -1, 1, 6, -5, -10, 6, 4, -1, 0, 1, -1, -6, 5, 10, -6, -4, 1, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1, 0, -1, 1, 8, -7, -21, 15, 20, -10, -5, 1, -1, 1, 10, -9, -36, 28, 56

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• tabf
• sign

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.

This sequence uses the characteristic polynomial defined as det(A - x I), rather than det(x I - A), so the last term in row n is (-1)^n. - Robert Israel, Feb 10 2023

Examples

			Top of the array:
   1, -1;
   0, -1,  1;
  -1,  1,  2, -1;
   0,  1, -1, -2, 1;
		

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)

Crossrefs

Cf. A204171, A202605, A204016.

Programs

Maple

for n from 1 to 20 do
  P:= (-1)^n * LinearAlgebra:-CharacteristicPolynomial(Matrix(n,n,(i,j) -> max(i,j) mod 2),x):
  print(seq(coeff(P,x,i),i=0..n));
od: # Robert Israel, Feb 10 2023

Mathematica

f[i_, j_] := If[Mod[Max[i, j], 2] == 1, 1, 0]
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}]]  (* A204171 *)
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[%]                 (* A204172 *)
TableForm[Table[c[n], {n, 1, 10}]]

Formula

(Empirical) T(m,k) = [x^m y^(k-1)] y*(1-x*y)*(1-x+x^3*y^2)/(1+y^2-2*x^2*y^2+x^4*y^4). - Robert Israel, Feb 10 2023