%I #6 Jul 12 2012 00:39:59

%S 1,-1,2,-4,1,7,-21,9,-1,34,-146,83,-16,1,201,-1277,878,-226,25,-1,

%T 1266,-13504,10729,-3340,500,-36,1,6063,-167689,149971,-53679,9805,

%U -967,49,-1,-44190,-2392326,2368995,-946036,199829

%N 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.

%C 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.

%e (For references regarding interlacing roots, see A202605.)

%e Top of the array:

%e 1....-1

%e 2....-4.....1

%e 7....-21....9....-1

%e 34...-146...83...-16...1

%t f[i_, j_] := 0; f[1, j_] := 1;

%t f[i_, 1] := 1; f[i_, i_] := 2 i - 1;

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[8]] (* 8x8 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t {n, 1, 15}, {i, 1, n}]] (* A204181 *)

%t p[n_] := CharacteristicPolynomial[m[n], x];

%t c[n_] := CoefficientList[p[n], x]

%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%] (* A204182 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A204181, A202605, A204016.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 12 2012