A204134 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1 (as in A204131).

1, -1, 1, -3, 1, 3, -11, 7, -1, 21, -83, 64, -15, 1, 315, -1287, 1074, -300, 31, -1, 9765, -40527, 35067, -10570, 1287, -63, 1, 615195, -2572731, 2265129, -707539, 92653, -5313, 127, -1, 78129765, -327967227, 291222882, -92551369

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
3....-11....7....-1
21...-83....64...-15...1
		

References

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

Crossrefs

Cf. A204133, A202605, A204016.

Programs

Mathematica

f[i_, j_] := 1; f[i_, i_] := 2^(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}]]  (* A204133 *)
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[%]                 (* A204134 *)
TableForm[Table[c[n], {n, 1, 10}]]