A204007 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j-2,2j+i-2} (A204006).

1, -1, 0, -5, 1, -1, -1, 12, -1, -2, 7, 5, -22, 1, -3, 19, -28, -15, 35, -1, -4, 35, -99, 84, 35, -51, 1, -5, 55, -220, 375, -210, -70, 70, -1, -6, 79, -403, 990, -1155, 462, 126, -92, 1, -7, 107, -660, 2093, -3575, 3069, -924, -210, 117, -1

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
 0....-5....1
-1....-1....12....-1
-2.....7....5.....-22...1
		

References

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

Crossrefs

Cf. A204006, A202605.

Programs

Mathematica

f[i_, j_] := Min[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}]]   (* A204006 *)
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[%]                (* A204007 *)
TableForm[Table[c[n], {n, 1, 10}]]