A204029 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=min(3i-2,3j-2) (A204028).

1, -1, 3, -5, 1, 9, -21, 12, -1, 27, -81, 75, -22, 1, 81, -297, 378, -195, 35, -1, 243, -1053, 1701, -1260, 420, -51, 1, 729, -3645, 7128, -6885, 3402, -798, 70, -1, 2187, -12393, 28431, -33858, 22275, -7938, 1386, -92, 1, 6561

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.

Examples

			Top of the array:
1....-1
3....-5....1
9....-21...12...-1
27...-81...75...-22....-11
		

References

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

Crossrefs

Cf. A204028, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Min[3 i - 2, 3 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}]]  (* A204028 *)
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[%]                 (* A204029 *)
TableForm[Table[c[n], {n, 1, 10}]]