A204182 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)=2i-1; f(i,j)=0 otherwise; as in A204181.

1, -1, 2, -4, 1, 7, -21, 9, -1, 34, -146, 83, -16, 1, 201, -1277, 878, -226, 25, -1, 1266, -13504, 10729, -3340, 500, -36, 1, 6063, -167689, 149971, -53679, 9805, -967, 49, -1, -44190, -2392326, 2368995, -946036, 199829

Offset: 1

Author

Clark Kimberling, Jan 12 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 real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

Examples

			(For references regarding interlacing roots, see A202605.)
Top of the array:
1....-1
2....-4.....1
7....-21....9....-1
34...-146...83...-16...1
		

Crossrefs

Cf. A204181, A202605, A204016.

Programs

Mathematica

f[i_, j_] := 0; f[1, j_] := 1;
f[i_, 1] := 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}]]  (* A204181 *)
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[%]                 (* A204182 *)
TableForm[Table[c[n], {n, 1, 10}]]