A203956 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203955.

1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -27, 165, -35, 1, 1, -123, 1255, -511, 54, -1, 1, -300, 9266, -6003, 1197, -82, 1, 1, -558, 77523, -71564, 20779, -2463, 111, -1, 1, -2841, 688624, -817771, 315489, -54393, 4386, -144, 1, 1, -9093

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...-6....1
1...-12....20....-1
1...-27....165...-35....1
1...-123...1255..-511...54...-1
		

References

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

Crossrefs

Cf. A203955, A202605.

Programs

Mathematica

t = {1, 2, 3}; 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}]
Flatten[%]  (* A203956 *)
TableForm[Table[c[n], {n, 1, 10}]]