A203950 - OEIS

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

%S 1,-1,1,-3,1,1,-6,5,-1,1,-13,18,-8,1,1,-24,52,-40,12,-1,1,-39,131,

%T -155,78,-16,1,1,-58,291,-508,391,-138,21,-1,1,-81,584,-1410,1548,

%U -840,225,-27,1,1,-108,1078,-3448,5151,-4016,1658,-348,33,-1,1

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

%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...-3....1

%e 1...-6....5....-1

%e 1...-13....18...-8....1

%e 1...-24...52...-40...12...-1

%t t = {1, 1, 0}; t1 = Flatten[{t, t, 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}]  (* A203950 *)

%t Flatten[%]

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

%Y Cf. A203949, A202605.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 08 2012