A203995 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i-j+1,j-i+1} (A203994).

1, -1, 1, -2, 1, 0, -2, 3, -1, -4, 8, 0, -4, 1, -16, 56, -56, 10, 5, -1, -48, 224, -360, 224, -35, -6, 1, -128, 736, -1584, 1560, -672, 84, 7, -1, -320, 2176, -5824, 7744, -5280, 1680, -168, -8, 1, -768, 6016, -19200, 32032, -29744

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...-2....1
 0...-2....3...-1
-4....8....0...-4....1
		

References

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

Crossrefs

Cf. A203994, 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}]]  (* A203994 *)
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[%]    (* A203995 *)
TableForm[Table[c[n], {n, 1, 10}]]