A204001 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1-1),j(i+1)-1} (A204000).

1, -1, 1, -6, 1, 1, -9, 17, -1, 1, -12, 39, -36, 1, 1, -15, 69, -119, 65, -1, 1, -18, 107, -272, 294, -106, 1, 1, -21, 153, -515, 846, -630, 161, -1, 1, -24, 207, -868, 1925, -2232, 1218, -232, 1, 1, -27, 269, -1351, 3783, -6017, 5214

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
1...-6....1
1...-9....17...-1
1...-12...39...-36...1
		

References

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

Crossrefs

Cf. A204000, A202605.

Programs

Mathematica

f[i_, j_] := Min[i (j + 1) - 1, j (i + 1) - 1];
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}]]  (* A204000 *)
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[%]               (* A204001 *)
TableForm[Table[c[n], {n, 1, 10}]]