A203952 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203949.

1, -1, 1, -2, 1, 1, -3, 3, -1, 1, -4, 6, -4, 1, 1, -6, 13, -13, 6, -1, 1, -8, 24, -34, 24, -8, 1, 1, -10, 39, -75, 75, -39, 10, -1, 1, -12, 58, -144, 195, -144, 58, -12, 1, 1, -14, 81, -250, 444, -459, 271, -89, 15, -1, 1, -16, 108, -400, 886

Offset: 1

Author

Clark Kimberling, Jan 08 2012

Keywords

• tabf
• 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. A203951, 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}]]