A204183 Symmetric matrix based on f(i,j) defined by f(i,1)=f(1,j)=1; f(i,i)= (-1)^(i-1); f(i,j)=0 otherwise; by antidiagonals.

1, 1, 1, 1, -1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0

Offset

1

Comments

A204183 represents the matrix M given by f(i,j) for i>=1 and j>=1. See A204184 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

Links

Table of n, a(n) for n=1..93.

Example

Northwest corner:
1...1...1...1...1...1
1..-1...0...0...0...0
1...0...1...0...0...0
1...0...0..-1...0...0
1...0...0...0...1...0

Mathematica

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1;
f[i_, i_] := (-1)^(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}]]  (* A204183 *)
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[%]                 (* A204184 *)
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A204184, A204016, A202453.

Sequence in context: A358672 A308016 A166360 * A204177 A185917 A143104

Adjacent sequences: A204180 A204181 A204182 * A204184 A204185 A204186

Keyword

sign,tabl

Author

Clark Kimberling, Jan 12 2012

Status

approved