A204024 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i(i+1)/2, j(j+1)/2) (A106255).

1, -1, 2, -4, 1, 6, -16, 10, -1, 24, -76, 70, -20, 1, 120, -428, 496, -224, 35, -1, 720, -2808, 3808, -2260, 588, -56, 1, 5040, -21096, 32152, -23008, 8140, -1344, 84, -1, 40320, -178848, 298688, -245560, 107328, -24772, 2772

Offset: 1

Author

Clark Kimberling, Jan 11 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
2....-4....1
6....-16...10...-1
24...-76...70...-20....1
		

References

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

Crossrefs

Cf. A106255, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Min[i (i + 1)/2, j (j + 1)/2];
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, 15}, {i, 1, n}]]     (* A106255 *)
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[%]                    (* A204024 *)
TableForm[Table[c[n], {n, 1, 10}]]