A204133 Symmetric matrix based on f(i,j)=(2^(i-1) if i=j and 1 otherwise), by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Clark Kimberling, Jan 11 2012

Keywords

• nonn
• tabl

Comments

A204133 represents the matrix M given by f(i,j)=(2^(i-1) if i=j and 1 otherwise) for i>=1 and j>=1. See A204134 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

Examples

			Northwest corner:
1 1 1 1 1
1 2 1 1 1
1 1 4 1 1
1 1 1 6 1
1 1 1 1 8
		

Crossrefs

Cf. A204134, A204016, A202453.

Programs

Mathematica

f[i_, j_] := 1; f[i_, i_] := 2^(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}]]  (* A204133 *)
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[%]                 (* A204134 *)
TableForm[Table[c[n], {n, 1, 10}]]