A204184 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)=(-1)^(i-1); f(i,j)=0 otherwise; as in A204181.

1, -1, -2, 0, 1, -1, 3, 1, -1, 2, -2, -5, 0, 1, 1, -5, -2, 6, 1, -1, -2, 4, 9, -4, -8, 0, 1, -1, 7, 3, -15, -3, 9, 1, -1, 2, -6, -13, 12, 21, -6, -11, 0, 1, 1, -9, -4, 28, 6, -30, -4, 12, 1, -1, -2, 8, 17, -24, -40, 24, 38, -8, -14, 0, 1, -1, 11, 5

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:
 1..-1
 2...0...1
-1...3...1..-1
 2..-2..-5...0..1
		

References

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

Crossrefs

Cf. A204183, A202605, A204016.

Programs

Mathematica

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1;
f[i_, i_] := (-1)^(i - 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}]]  (* A204183 *)
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[%]                 (* A204184 *)
TableForm[Table[c[n], {n, 1, 10}]]