A204135 - OEIS

%I #6 Jul 12 2012 00:39:58

%S 1,-1,2,-4,1,8,-28,17,-1,64,-384,424,-80,1,1024,-10624,19400,-7700,

%T 401,-1,32768,-598016,1748224,-1225536,161618,-2084,1,2097152,

%U -68550656,319410176,-363159040,95891872

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Delannoy matrix (A008288).

%C Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are real, and they interlace the zeros of p(n+1).  See A202605 and A204016 for guides to related sequences.

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

%e Top of the array:

%e 1....-1

%e 2....-4.....1

%e 8....-28....17....-1

%e 64...-384...424...-80...1

%e The interlacing of zeros is illustrated by these zeros (truncated):

%e p(1):  1

%e p(2): .58, 3.41

%e p(3): .36, 1.44, 15.19

%e p(4): .21, .87, 4.53, 74.3

%e p(5): .12, .59, 2.14, 17.22, 380.91

%t f[i_, 1] := 1; f[1, j_] := 1;

%t f[i_, j_] := f[i, j - 1] + f[i - 1, j - 1] + f[i - 1, j]

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[8]] (* 8x8 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t   {n, 1, 15}, {i, 1, n}]]  (*  Delannoy, A008288 *)

%t p[n_] := CharacteristicPolynomial[m[n], x];

%t c[n_] := CoefficientList[p[n], x]

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

%t Table[c[n], {n, 1, 8}]

%t Flatten[%]                 (* 204135 *)

%t TableForm[Table[c[n], {n, 1, 6}]]

%Y Cf. A008288, A202605, A204016.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 12 2012