A204001 - OEIS

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

%S 1,-1,1,-6,1,1,-9,17,-1,1,-12,39,-36,1,1,-15,69,-119,65,-1,1,-18,107,

%T -272,294,-106,1,1,-21,153,-515,846,-630,161,-1,1,-24,207,-868,1925,

%U -2232,1218,-232,1,1,-27,269,-1351,3783,-6017,5214

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1-1),j(i+1)-1} (A204000).

%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 1...-9....17...-1

%e 1...-12...39...-36...1

%t f[i_, j_] := Min[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}]]  (* A204000 *)

%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[%]               (* A204001 *)

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

%Y Cf. A204000, A202605.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 09 2012