A204159 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(3i-2j, 3j-2i), as in A204158.

1, -1, -14, -3, 1, 115, 79, 6, -1, -800, -895, -255, -10, 1, 5125, 7875, 3850, 625, 15, -1, -31250, -60875, -42075, -12180, -1295, -21, 1, 184375, 434375, 387750, 162375, 31710, 2394, 28, -1, -1062500, -2934375

Offset: 1

Author

Clark Kimberling, Jan 12 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
-14....-3......1
 115....79.....6.....-1
-800...-895...-255...-10....1
		

References

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

Crossrefs

Cf. A204158, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Max[3 i - 2 j, 3 j - 2 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}]]   (* A204158 *)
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[%]                 (* A204159 *)
TableForm[Table[c[n], {n, 1, 10}]]