A204161 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (3i-2 if i=j and = 0 otherwise), as in A204160.

1, -1, 3, -5, 1, 18, -36, 12, -1, 162, -360, 153, -22, 1, 1944, -4644, 2295, -435, 35, -1, 29160, -73548, 40419, -9135, 990, -51, 1, 524880, -1382184, 823284, -210924, 27720, -1953, 70, -1, 11022480

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
3.....-5.....1
18....-36....12....-1
162...-360...153...-22...1
		

References

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

Crossrefs

Cf. A204160, 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}]]    (* A204160 *)
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[%]                   (* A204161 *)
TableForm[Table[c[n], {n, 1, 10}]]