A203948 - OEIS

%I #10 Jul 12 2012 00:39:54

%S 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 61,-14,1,1,-22,147,-308,275,-111,19,-1,1,-29,256,-763,1001,-629,186,

%U -24,1,1,-37,428,-1683,3013,-2721,1264,-291,30,-1,1,-46

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

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

%e Top of the array:

%e 1...-1

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

%e 1...-4....4....-1

%e 1...-7....13...-7....1

%e 1...-11...35...-31...10...-1

%t t = {1, 0, 1}; t1 = Flatten[{t, t, t, t, t, t, t}];

%t f[k_] := t1[[k]];

%t U[n_] :=

%t   NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[

%t    Table[f[k], {k, 1, n}]];

%t L[n_] := Transpose[U[n]];

%t p[n_] := CharacteristicPolynomial[L[n].U[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[%]

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

%Y Cf. A203947, A202605.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 08 2012