A204023 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-1, 2j-1) (A204022).

1, -1, -6, -4, 1, 20, 36, 9, -1, -56, -160, -120, -16, 1, 144, 560, 700, 300, 25, -1, -352, -1728, -3024, -2240, -630, -36, 1, 832, 4928, 11088, 11760, 5880, 1176, 49, -1, -1920, -13312, -36608, -50688, -36960

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
-6....-4.....1
 20....36....9.....-1
-56...-160..-120...-16....1
		

References

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

Crossrefs

Cf. A204022, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Max[2 i - 1, 2 j - 1];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204022 *)
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[%]                 (* A204023 *)
TableForm[Table[c[n], {n, 1, 10}]]