A204144 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=max(ceiling(i/j),ceiling(j/i)) (as in A204143).

1, -1, -3, -2, 1, 8, 14, 3, -1, -12, -42, -35, -4, 1, 19, 95, 145, 73, 5, -1, -20, -140, -338, -336, -125, -6, 1, 16, 184, 665, 1037, 735, 205, 7, -1, -16, -212, -981, -2140, -2381, -1320, -303, -8, 1, 12, 200, 1209, 3581, 5727, 5021

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
-3...-2....1
 8....14...3....-1
-12..-42..-35...-4....1
		

References

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

Crossrefs

Cf. A204143, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Max[Ceiling[i/j], Ceiling[j/i]];
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}]]  (* A204143 *)
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[%]                 (* A204144 *)
TableForm[Table[c[n], {n, 1, 10}]]