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A203952 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203949.

%I A203952 #14 Feb 21 2023 07:34:25
%S A203952 1,-1,1,-2,1,1,-3,3,-1,1,-4,6,-4,1,1,-6,13,-13,6,-1,1,-8,24,-34,24,-8,
%T A203952 1,1,-10,39,-75,75,-39,10,-1,1,-12,58,-144,195,-144,58,-12,1,1,-14,81,
%U A203952 -250,444,-459,271,-89,15,-1,1,-16,108,-400,886
%N A203952 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203949.
%C A203952 Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are positive, and they interlace the zeros of p(n+1).  See A202605 for a guide to related sequences.
%D A203952 (For references regarding interlacing roots, see A202605.)
%e A203952 Top of the array:
%e A203952 1...-1
%e A203952 1...-3....1
%e A203952 1...-6....5....-1
%e A203952 1...-13...18...-8....1
%e A203952 1...-24...52...-40...12...-1
%t A203952 t = {1, 1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t}];
%t A203952 f[k_] := t1[[k]];
%t A203952 U[n_] :=
%t A203952   NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
%t A203952    Table[f[k], {k, 1, n}]];
%t A203952 L[n_] := Transpose[U[n]];
%t A203952 p[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A203952 c[n_] := CoefficientList[p[n], x]
%t A203952 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203952 Table[c[n], {n, 1, 12}]  (* A203950 *)
%t A203952 Flatten[%]
%t A203952 TableForm[Table[c[n], {n, 1, 10}]]
%Y A203952 Cf. A203951, A202605.
%K A203952 tabf,sign
%O A203952 1,4
%A A203952 _Clark Kimberling_, Jan 08 2012