A203999 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{i(j+1-1),j(i+1)-1} (A203998).

1, -1, -4, -6, 1, 7, 27, 17, -1, -10, -60, -99, -36, 1, 13, 105, 279, 269, 65, -1, -16, -162, -593, -944, -609, -106, 1, 19, 231, 1077, 2405, 2610, 1218, 161, -1, -22, -312, -1767, -5092, -7865, -6264, -2226, -232, 1, 25, 405, 2699, 9541

Offset: 1

Author

Clark Kimberling, Jan 09 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
-4....-6.....1
7.... 27....17...-1
-10...-60...-99...-36...1
		

References

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

Crossrefs

Cf. A203998, A202605.

Programs

Mathematica

f[i_, j_] := Max[i (j + 1) - 1, j (i + 1) - 1];
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}]]    (* A203998 *)
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[%]   (* A203999 *)
TableForm[Table[c[n], {n, 1, 10}]]