A203999 - OEIS

%I #7 Jul 12 2012 00:39:54

%S 1,-1,-4,-6,1,7,27,17,-1,-10,-60,-99,-36,1,13,105,279,269,65,-1,-16,

%T -162,-593,-944,-609,-106,1,19,231,1077,2405,2610,1218,161,-1,-22,

%U -312,-1767,-5092,-7865,-6264,-2226,-232,1,25,405,2699,9541

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{i(j+1-1),j(i+1)-1} (A203998).

%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 -4....-6.....1

%e 7.... 27....17...-1

%e -10...-60...-99...-36...1

%t f[i_, j_] := Max[i (j + 1) - 1, j (i + 1) - 1];

%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}]]    (* A203998 *)

%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[%]   (* A203999 *)

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

%Y Cf. A203998, A202605.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 09 2012