A204003 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j,i+2j} (A204002).

3, -1, 2, -9, 1, 1, -9, 18, -1, 0, -5, 25, -30, 1, -1, 3, 14, -55, 45, -1, -2, 15, -27, -28, 105, -63, 1, -3, 31, -110, 135, 42, -182, 84, -1, -4, 51, -247, 550, -495, -42, 294, -108, 1, -5, 75, -450, 1365, -2145, 1485, 0, -450, 135, -1, -6, 103

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:
3...-1
2...-9.....1
1...-9....18...-1
0...-5....25...-30...1
		

References

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

Crossrefs

Cf. A204002, A202605.

Programs

Mathematica

f[i_, j_] := Min[2 i + j, 2 j + i];
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}]]    (* A204002 *)
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[%]                 (* A204003 *)
TableForm[Table[c[n], {n, 1, 10}]]