A203997 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1),j(i+1)} (A203996).

2, -1, 3, -8, 1, 4, -19, 20, -1, 5, -34, 69, -40, 1, 6, -53, 160, -189, 70, -1, 7, -76, 305, -552, 434, -112, 1, 8, -103, 516, -1265, 1560, -882, 168, -1, 9, -134, 805, -2496, 4235, -3828, 1638, -240, 1, 10, -169, 1184, -4445, 9646

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:
2...-1
3...-8.....1
4...-19....20....-1
5...-34....69....-40....1
6...-53....160...-189...70....-1
7...-76....305...-552...434...-112...1
		

References

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

Crossrefs

Cf. A203996, A202605.

Programs

Mathematica

f[i_, j_] := Min[i (j + 1), j (i + 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}]]   (* A203996 *)
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[%]      (* A203997 *)
TableForm[Table[c[n], {n, 1, 10}]]