A204170 Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of (i*j), as in A003991.

1, -1, 0, -5, 1, 0, 0, 14, -1, 0, 0, 0, -30, 1, 0, 0, 0, 0, 55, -1, 0, 0, 0, 0, 0, -91, 1, 0, 0, 0, 0, 0, 0, 140, -1, 0, 0, 0, 0, 0, 0, 0, -204, 1, 0, 0, 0, 0, 0, 0, 0, 0, 285, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -385, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 506, -1

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.

p(n,x) = x^n + (-1)^n*s(n)*x^n - 1, where s=A000330 (square pyramidal numbers).

Examples

			Top of the array:
  1, -1;
  0, -5,  1;
  0,  0, 14,  -1;
  0,  0,  0, -30, 1;
		

References

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

Crossrefs

Cf. A003991, A202605, A204016.

Programs

Mathematica

f[i_, j_] := i*j;
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}]]  (* A003991 *)
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[%]                (* A204170 *)
TableForm[Table[c[n], {n, 1, 10}]]