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A204157 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.

%I A204157 #6 Jul 12 2012 00:39:58
%S A204157 1,-1,-13,-4,1,88,78,9,-1,-496,-704,-260,-16,1,2560,4960,3080,650,25,
%T A204157 -1,-12544,-30720,-26784,-9856,-1365,-36,1,59392,175616,197120,104160,
%U A204157 25872,2548,49,-1,-274432,-950272,-1304576,-901120,-327360,-59136,-4368,-64,1,1245184,4939776,8017920,6849024,3294720
%N A204157 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 A204157 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 A204157 (For references regarding interlacing roots, see A202605.)
%e A204157 Top of the array:
%e A204157  1....-1
%e A204157 -13...-4.....1
%e A204157  88....78....9.....-1
%e A204157 -496..-704..-260...-16...1
%t A204157 f[i_, j_] := -1 + Max[3 i - j, 3 j - i];
%t A204157 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204157 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204157 Flatten[Table[f[i, n + 1 - i],
%t A204157   {n, 1, 15}, {i, 1, n}]]    (* A204156 *)
%t A204157 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204157 c[n_] := CoefficientList[p[n], x]
%t A204157 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204157 Table[c[n], {n, 1, 12}]
%t A204157 Flatten[%]                   (* A204157 *)
%t A204157 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204157 Cf. A204156, A202605, A204016.
%K A204157 tabl,sign
%O A204157 1,3
%A A204157 _Clark Kimberling_, Jan 12 2012