A204157 - OEIS

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

%S 1,-1,-13,-4,1,88,78,9,-1,-496,-704,-260,-16,1,2560,4960,3080,650,25,

%T -1,-12544,-30720,-26784,-9856,-1365,-36,1,59392,175616,197120,104160,

%U 25872,2548,49,-1,-274432,-950272,-1304576,-901120,-327360,-59136,-4368,-64,1,1245184,4939776,8017920,6849024,3294720

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(3i-j, 3j-i), as in A204156.

%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 -13...-4.....1

%e  88....78....9.....-1

%e -496..-704..-260...-16...1

%t f[i_, j_] := -1 + Max[3 i - j, 3 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}]]    (* A204156 *)

%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[%]                   (* A204157 *)

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

%Y Cf. A204156, A202605, A204016.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 12 2012