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A203948 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203947.

%I A203948 #10 Jul 12 2012 00:39:54
%S A203948 1,-1,1,-2,1,1,-4,4,-1,1,-7,13,-7,1,1,-11,35,-31,10,-1,1,-16,74,-107,
%T A203948 61,-14,1,1,-22,147,-308,275,-111,19,-1,1,-29,256,-763,1001,-629,186,
%U A203948 -24,1,1,-37,428,-1683,3013,-2721,1264,-291,30,-1,1,-46
%N A203948 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203947.
%C A203948 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 A203948 (For references regarding interlacing roots, see A202605.)
%e A203948 Top of the array:
%e A203948 1...-1
%e A203948 1...-2....1
%e A203948 1...-4....4....-1
%e A203948 1...-7....13...-7....1
%e A203948 1...-11...35...-31...10...-1
%t A203948 t = {1, 0, 1}; t1 = Flatten[{t, t, t, t, t, t, t}];
%t A203948 f[k_] := t1[[k]];
%t A203948 U[n_] :=
%t A203948   NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
%t A203948    Table[f[k], {k, 1, n}]];
%t A203948 L[n_] := Transpose[U[n]];
%t A203948 p[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A203948 c[n_] := CoefficientList[p[n], x]
%t A203948 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203948 Table[c[n], {n, 1, 12}]
%t A203948 Flatten[%]
%t A203948 TableForm[Table[c[n], {n, 1, 10}]]
%Y A203948 Cf. A203947, A202605.
%K A203948 tabl,sign
%O A203948 1,4
%A A203948 _Clark Kimberling_, Jan 08 2012