A203946 - OEIS

%I #14 Feb 13 2023 05:25:33

%S 1,-1,1,-2,1,1,-3,3,-1,1,-5,8,-5,1,1,-7,17,-17,7,-1,1,-9,30,-45,30,-9,

%T 1,1,-11,47,-98,103,-52,12,-1,1,-13,68,-183,269,-212,83,-15,1,1,-15,

%U 93,-308,588,-651,399,-123,18,-1,1,-17,122,-481,1136

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203945.

%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...-3....3....-1

%e 1...-5....8....-5....1

%e 1...-7....17...-17...7...-1

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

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

%t U[n_] := NestList[Most[Prepend[#, 0]] &, #,

%t Length[#] - 1] &[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[%]   (* A203946 *)

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

%Y Cf. A203945, A202605.

%K tabf,sign

%O 1,4

%A _Clark Kimberling_, Jan 08 2012