A204013 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{3i+j-3,i+3j-3} (A204012).

1, -1, 1, -6, 1, 0, -10, 15, -1, -4, -8, 40, -28, 1, -16, 24, 56, -110, 45, -1, -48, 160, -72, -224, 245, -66, 1, -128, 608, -880, 120, 672, -476, 91, -1, -320, 1920, -4160, 3520, 0, -1680, 840, -120, 1, -768, 5504, -15360, 20384

Offset: 1

Author

Clark Kimberling, Jan 10 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
 1....-6....1
 0....-10...15....-1
-4....-8....40....-28....1
		

References

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

Crossrefs

Cf. A204012, A202605.

Programs

Mathematica

f[i_, j_] := Min[3 i + j - 3, 3 j + i - 3];
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}]]   (* A204012 *)
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[%]  (* A204013 *)
TableForm[Table[c[n], {n, 1, 10}]]