A203991 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {(i+j)*min(i,j)} (A203990).

2, -1, 7, -10, 1, 38, -71, 28, -1, 281, -610, 357, -60, 1, 2634, -6329, 4620, -1253, 110, -1, 29919, -77530, 65613, -23348, 3514, -182, 1, 399342, -1098271, 1036044, -442349, 90800, -8442, 280, -1, 6125265

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
7.... -10... 1
38... -71... 28... -1
281.. -610.. 357.. -60... 1
		

References

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

Crossrefs

Cf. A203990, A202605.

Programs

Mathematica

f[i_, j_] := (i + j) Min[i, j];
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}]]  (* A203990 *)
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[%]               (* A203991 *)
TableForm[Table[c[n], {n, 1, 10}]]