A203950 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203949.

1, -1, 1, -3, 1, 1, -6, 5, -1, 1, -13, 18, -8, 1, 1, -24, 52, -40, 12, -1, 1, -39, 131, -155, 78, -16, 1, 1, -58, 291, -508, 391, -138, 21, -1, 1, -81, 584, -1410, 1548, -840, 225, -27, 1, 1, -108, 1078, -3448, 5151, -4016, 1658, -348, 33, -1, 1

Offset: 1

Author

Clark Kimberling, Jan 08 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 positive, and they interlace the zeros of p(n+1). See A202605 for a guide to related sequences.

Examples

			Top of the array:
1...-1
1...-3....1
1...-6....5....-1
1...-13....18...-8....1
1...-24...52...-40...12...-1
		

References

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

Crossrefs

Cf. A203949, A202605.

Programs

Mathematica

t = {1, 1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t}];
f[k_] := t1[[k]];
U[n_] :=
  NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
p[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]  (* A203950 *)
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]