A204167 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of ceiling((i+j)/2), as in A204166.

1, -1, -2, -3, 1, 1, 6, 6, -1, 0, -4, -16, -10, 1, 0, 0, 15, 32, 15, -1, 0, 0, 0, -36, -60, -21, 1, 0, 0, 0, 0, 84, 100, 28, -1, 0, 0, 0, 0, 0, -160, -160, -36, 1, 0, 0, 0, 0, 0, 0, 300, 240, 45, -1, 0, 0, 0, 0, 0, 0, 0, -500, -350

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• tabf
• 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,  -3,   1
   1,   6,   6,  -1
   0,  -4, -16, -10,   1
		

References

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

Crossrefs

Cf. A204166, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Ceiling[(i + j)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204166 *)
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[%]                 (* A204167 *)
TableForm[Table[c[n], {n, 1, 10}]]

Extensions

Definition corrected by Georg Fischer, Nov 29 2021