A203992 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (A143182 in square format).

1, -1, -3, -2, 1, 8, 14, 3, -1, -20, -56, -40, -4, 1, 48, 184, 224, 90, 5, -1, -112, -544, -936, -672, -175, -6, 1, 256, 1504, 3344, 3480, 1680, 308, 7, -1, -576, -3968, -10816, -14784, -10560, -3696, -504, -8, 1, 1280, 10112, 32640, 55328

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
-3... -1.... 1
 8.... 14... 3... -1
-20.. -56.. -40.. -4... 1
		

References

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

Crossrefs

Cf. A143182, A202605.

Programs

Mathematica

f[i_, j_] := Max[i - j + 1, j - i + 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}]]  (* A143182 in square format *)
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[%]    (* A203992 *)
TableForm[Table[c[n], {n, 1, 10}]]