A204020 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i,j)^2 (A106314).

1, -1, 3, -5, 1, 15, -31, 14, -1, 105, -247, 157, -30, 1, 945, -2433, 1892, -553, 55, -1, 10395, -28653, 25573, -9620, 1554, -91, 1, 135135, -393279, 388810, -173773, 37550, -3738, 140, -1, 2027025, -6169455

Offset: 1

Author

Clark Kimberling, Jan 11 2012

Keywords

• tabl
• 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.

Constant term of p(n,x) is A001147(n), and the coefficient of the linear term is A000330(n). - Enrique Pérez Herrero, Feb 20 2013

Examples

			Top of the array:
1.....-1
3.....-5.....1
15....-31....14....-1
105...-247...157...-30...1
		

References

• (For references regarding interlacing roots, see A202605.)

Crossrefs

Cf. A106314, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Min[i^2, j^2];
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, 15}, {i, 1, n}]]   (* A106314 *)
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[%]                  (* A204020 *)
TableForm[Table[c[n], {n, 1, 10}]]