A203954 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203953.

1, -1, 1, -6, 1, 1, -20, 12, -1, 1, -70, 75, -22, 1, 1, -264, 406, -200, 33, -1, 1, -1034, 2085, -1470, 430, -48, 1, 1, -4108, 10296, -9600, 4116, -816, 64, -1, 1, -16398, 49231, -57574, 33135, -9786, 1410, -84, 1, 1, -65552, 229482

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...-20....12....-1
1...-70....75....-22....1
1...-264...406...-200...33...-1
		

References

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

Crossrefs

Cf. A203953, A202605.

Programs

Mathematica

t = {1, 2}; 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[%]  (* A203954 *)
TableForm[Table[c[n], {n, 1, 10}]]