A204144 - OEIS

%I #6 Jul 12 2012 00:39:58

%S 1,-1,-3,-2,1,8,14,3,-1,-12,-42,-35,-4,1,19,95,145,73,5,-1,-20,-140,

%T -338,-336,-125,-6,1,16,184,665,1037,735,205,7,-1,-16,-212,-981,-2140,

%U -2381,-1320,-303,-8,1,12,200,1209,3581,5727,5021

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=max(ceiling(i/j),ceiling(j/i)) (as in A204143).

%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.

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

%e Top of the array:

%e  1...-1

%e -3...-2....1

%e  8....14...3....-1

%e -12..-42..-35...-4....1

%t f[i_, j_] := Max[Ceiling[i/j], Ceiling[j/i]];

%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}]]  (* A204143 *)

%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[%]                 (* A204144 *)

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

%Y Cf. A204143, A202605, A204016.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 11 2012