A203948 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203947.

1, -1, 1, -2, 1, 1, -4, 4, -1, 1, -7, 13, -7, 1, 1, -11, 35, -31, 10, -1, 1, -16, 74, -107, 61, -14, 1, 1, -22, 147, -308, 275, -111, 19, -1, 1, -29, 256, -763, 1001, -629, 186, -24, 1, 1, -37, 428, -1683, 3013, -2721, 1264, -291, 30, -1, 1, -46

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...-2....1
1...-4....4....-1
1...-7....13...-7....1
1...-11...35...-31...10...-1
		

References

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

Crossrefs

Cf. A203947, A202605.

Programs

Mathematica

t = {1, 0, 1}; t1 = Flatten[{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}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]