A204178 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if i=1 or j=1 or i=j, and 0 otherwise) as in A204177.

1, -1, 0, -2, 1, -1, -1, 3, -1, -2, 2, 3, -4, 1, -3, 7, -2, -6, 5, -1, -4, 14, -15, 0, 10, -6, 1, -5, 23, -39, 25, 5, -15, 7, -1, -6, 34, -77, 84, -35, -14, 21, -8, 1, -7, 47, -132, 196, -154, 42, 28, -28, 9, -1, -8, 62, -207, 384, -420, 252

Offset

1,4

Comments

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

References

(For references regarding interlacing roots, see A202605.)

Links

Table of n, a(n) for n=1..60.

Example

Top of the array:
1...-1
0...-2.....1
-1...-1.....3...-1
-2....2.....3...-4...1
-3....7....-2...-6...5...-1

Mathematica

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := 1;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204177 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                 (* A204178 *)
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A204177, A202605, A204016.

Sequence in context: A260221 A014671 A029365 * A365466 A095136 A105540

Adjacent sequences: A204175 A204176 A204177 * A204179 A204180 A204181

Keyword

tabf,sign

Author

Clark Kimberling, Jan 12 2012

Status

approved