A204017 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{j mod i, i mod j} (A204016).

0, -1, -1, 0, 1, 4, 6, 0, -1, -15, -38, -20, 0, 1, 56, 206, 184, 50, 0, -1, -185, -1072, -1357, -630, -105, 0, 1, 204, 5146, 9276, 6060, 1736, 196, 0, -1, 6209, -17334, -58470, -52452, -21102, -4116, -336, 0, 1, -112400, -67682, 293984

Offset: 1

Author

Clark Kimberling, Jan 10 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 A204016 and A202605 for guides to related sequences.

Examples

			Top of the array:
 1... -1
-1.... 0.... 1
 4.... 6.... 0... -1
-15.. -38.. -20... 0... 1
 56... 206.. 184.. 50.. 0.. -1
...
The 1st principal submatrix (ps) of A204016 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{0,1},{1,0}}, with p(2)=-1+x^2 and zero-set {-1,1}.
...
The 3rd ps is {{0,1,1},{1,0,2},{1,2,0}}, with p(3)=4+6x-x^3 and zero-set {-2, -0.732...,2.732...}.
...
The 4th ps is {{0,1,1,1},{1,0,2,2},{1,2,0,3},{1,2,0,3}}, with p(4)=-15-38x-20x^2+x^4 and zero-set {-3, -1.714, -0.553, 5.268}.
...
The interlace property is illustrated for the last two zero-sets by this chain:
-3 < -2 < -1.7 < -0.7 < -0.5 < 2.7 < 5.2
		

References

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

Crossrefs

Cf. A204016, A202605.

Programs

Mathematica

f[i_, j_] := Max[Mod[i, j], Mod[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, 12}, {i, 1, n}]]  (* A204016 *)
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[%]               (* A204017 *)
TableForm[Table[c[n], {n, 1, 10}]]