A204169 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j-1), as in A002024.

1, -1, -1, -4, 1, 0, 6, 9, -1, 0, 0, -20, -16, 1, 0, 0, 0, 50, 25, -1, 0, 0, 0, 0, -105, -36, 1, 0, 0, 0, 0, 0, 196, 49, -1, 0, 0, 0, 0, 0, 0, -336, -64, 1, 0, 0, 0, 0, 0, 0, 0, 540, 81, -1, 0, 0, 0, 0, 0, 0, 0, 0, -825, -100, 1

Offset: 1

Author

Clark Kimberling, Jan 12 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 and A204016 for guides to related sequences.

Examples

			Top of the array:
2....-1
-1....-4.....1
0.....6.....9....-1
0.....0....-20...-16...1
		

References

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

Crossrefs

Cf. A002024, A202605, A204016.

Programs

Mathematica

f[i_, j_] := i + j - 1;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A002024 *)
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[%]                (* A204169 *)
TableForm[Table[c[n], {n, 1, 10}]]