A204132 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, 2, -4, 1, 8, -20, 9, -1, 48, -136, 80, -16, 1, 384, -1184, 820, -220, 25, -1, 3840, -12608, 9784, -3160, 490, -36, 1, 46080, -158976, 134400, -49504, 9380, -952, 49, -1, 645120, -2317824, 2097024, -853440, 186704

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
2....-4.....1
8....-20....9...-1
48...-136...80..-16...1
		

References

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

Crossrefs

Cf. A204131, 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}]]  (* A204131 *)
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[%]                 (* A204132 *)
TableForm[Table[c[n], {n, 1, 10}]]