A204005 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{2i+j-2,2j+i-2} (A204004).

1, -1, -5, -5, 1, 9, 31, 12, -1, -13, -73, -105, -22, 1, 17, 131, 322, 265, 35, -1, -21, -205, -711, -1036, -560, -51, 1, 25, 295, 1320, 2775, 2730, 1050, 70, -1, -29, -401, -2197, -6050, -8745, -6258, -1806, -92, 1, 33, 523, 3390, 11557

Offset: 1

Author

Clark Kimberling, Jan 09 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 for a guide to related sequences.

Examples

			Top of the array:
1....-1
-5....-5....1
9.....31...12....-1
-13...-73..-105...-22...1
		

References

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

Crossrefs

Cf. A204004, A202605.

Programs

Mathematica

f[i_, j_] := Max[2 i + j - 2, 2 j + i - 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, 12}, {i, 1, n}]]  (* A204004 *)
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[%]   (* A204005 *)
TableForm[Table[c[n], {n, 1, 10}]]