A204013 - OEIS

%I #6 Jul 12 2012 00:39:54

%S 1,-1,1,-6,1,0,-10,15,-1,-4,-8,40,-28,1,-16,24,56,-110,45,-1,-48,160,

%T -72,-224,245,-66,1,-128,608,-880,120,672,-476,91,-1,-320,1920,-4160,

%U 3520,0,-1680,840,-120,1,-768,5504,-15360,20384

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{3i+j-3,i+3j-3} (A204012).

%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 for a guide to related sequences.

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

%e Top of the array:

%e  1....-1

%e  1....-6....1

%e  0....-10...15....-1

%e -4....-8....40....-28....1

%t f[i_, j_] := Min[3 i + j - 3, 3 j + i - 3];

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

%t TableForm[m[6]] (* 6x6 principal submatrix *)

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

%t {n, 1, 12}, {i, 1, n}]]   (* A204012 *)

%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[%]  (* A204013 *)

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

%Y Cf. A204012, A202605.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 10 2012