A204126 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

1, -1, 1, -3, 1, 2, -8, 6, -1, 6, -28, 29, -10, 1, 24, -124, 155, -75, 15, -1, 120, -668, 949, -565, 160, -21, 1, 720, -4248, 6636, -4564, 1610, -301, 28, -1, 5040, -31176, 52464, -40208, 16569, -3892, 518, -36, 1, 40320, -259488, 463956

Offset: 1

Author

Clark Kimberling, Jan 11 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:
1....-1
1....-3.....1
2....-8.....6....-1
6....-28....29...-10...1
		

References

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

Crossrefs

Cf. A204125, A202605, A204016.

Programs

Mathematica

f[i_, j_] := 1; f[i_, i_] := i;
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}]]  (* A204125 *)
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[%]                 (* A204126 *)
TableForm[Table[c[n], {n, 1, 10}]]