A204174 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(2i-1 if max(i,j) is odd, and 0 otherwise) as in A204173.

1, -1, 0, -1, 1, -4, 6, 3, -1, 0, 4, -6, -3, 1, 36, -60, -31, 33, 6, -1, 0, -36, 60, 31, -33, -6, 1, -576, 1008, 516, -736, -131, 105, 10, -1, 0, 576, -1008, -516, 736, 131, -105, -10, 1, 14400, -25920, -13116, 21628, 3621, -4581, -406, 255

Offset

1,6

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..52.

Example

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

Mathematica

f[i_, j_] := If[Mod[Max[i, j], 2] == 1,
(1 + Max[i, j])/2, 0]
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}]]  (* A204173 *)
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[%]                 (* A204174 *)
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A202605, A204016, A204173.

Sequence in context: A212084 A182368 A185442 * A086467 A197731 A325009

Adjacent sequences: A204171 A204172 A204173 * A204175 A204176 A204177

Keyword

tabf,sign

Author

Clark Kimberling, Jan 12 2012

Status

approved