A204015 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{1+(j mod i), 1+( i mod j)} (A204014).

1, -1, 0, -2, 1, -1, 3, 3, -1, 0, 2, -6, -4, 1, 0, -8, 8, 20, 5, -1, -16, 14, 58, -4, -31, -6, 1, 48, 16, -169, -121, 69, 63, 7, -1, 208, -320, -576, 540, 432, -128, -97, -8, 1, 400, -2048, 1876, 2340, -1828, -928, 309, 153, 9, -1, -4800, 6880

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 A202605 for a guide to related sequences.

Examples

			Top of the array:
 1...-1
 0...-2...1
-1....3...3...-1
 0....2..-6...-4....1
 0...-8...8....20...5...1
		

References

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

Crossrefs

Cf. A204014, A202605.

Programs

Mathematica

f[i_, j_] := Min[1 + Mod[i, j], 1 + Mod[j, i]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A204014 *)
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[%]   (* A204015 *)
TableForm[Table[c[n], {n, 1, 10}]]