A203946 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203945.

1, -1, 1, -2, 1, 1, -3, 3, -1, 1, -5, 8, -5, 1, 1, -7, 17, -17, 7, -1, 1, -9, 30, -45, 30, -9, 1, 1, -11, 47, -98, 103, -52, 12, -1, 1, -13, 68, -183, 269, -212, 83, -15, 1, 1, -15, 93, -308, 588, -651, 399, -123, 18, -1, 1, -17, 122, -481, 1136

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...-2....1
1...-3....3....-1
1...-5....8....-5....1
1...-7....17...-17...7...-1
		

References

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

Crossrefs

Cf. A203945, A202605.

Programs

Mathematica

t = {1, 0, 0}; t1 = Flatten[{t, 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}]
Flatten[%]   (* A203946 *)
TableForm[Table[c[n], {n, 1, 10}]]