cp's OEIS Frontend

A203992 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (A143182 in square format).

%I A203992 #6 Jul 12 2012 00:39:54
%S A203992 1,-1,-3,-2,1,8,14,3,-1,-20,-56,-40,-4,1,48,184,224,90,5,-1,-112,-544,
%T A203992 -936,-672,-175,-6,1,256,1504,3344,3480,1680,308,7,-1,-576,-3968,
%U A203992 -10816,-14784,-10560,-3696,-504,-8,1,1280,10112,32640,55328
%N A203992 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (A143182 in square format).
%C A203992 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 A203992 (For references regarding interlacing roots, see A202605.)
%e A203992 Top of the array:
%e A203992  1... -1
%e A203992 -3... -1.... 1
%e A203992  8.... 14... 3... -1
%e A203992 -20.. -56.. -40.. -4... 1
%t A203992 f[i_, j_] := Max[i - j + 1, j - i + 1];
%t A203992 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A203992 TableForm[m[6]]  (* 6x6 principal submatrix *)
%t A203992 Flatten[Table[f[i, n + 1 - i],
%t A203992 {n, 1, 12}, {i, 1, n}]]  (* A143182 in square format *)
%t A203992 p[n_] := CharacteristicPolynomial[m[n], x];
%t A203992 c[n_] := CoefficientList[p[n], x]
%t A203992 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203992 Table[c[n], {n, 1, 12}]
%t A203992 Flatten[%]    (* A203992 *)
%t A203992 TableForm[Table[c[n], {n, 1, 10}]]
%Y A203992 Cf. A143182, A202605.
%K A203992 tabl,sign
%O A203992 1,3
%A A203992 _Clark Kimberling_, Jan 09 2012