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A204135 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Delannoy matrix (A008288).

%I A204135 #6 Jul 12 2012 00:39:58
%S A204135 1,-1,2,-4,1,8,-28,17,-1,64,-384,424,-80,1,1024,-10624,19400,-7700,
%T A204135 401,-1,32768,-598016,1748224,-1225536,161618,-2084,1,2097152,
%U A204135 -68550656,319410176,-363159040,95891872
%N A204135 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Delannoy matrix (A008288).
%C A204135 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 A204135 (For references regarding interlacing roots, see A202605.)
%e A204135 Top of the array:
%e A204135 1....-1
%e A204135 2....-4.....1
%e A204135 8....-28....17....-1
%e A204135 64...-384...424...-80...1
%e A204135 The interlacing of zeros is illustrated by these zeros (truncated):
%e A204135 p(1):  1
%e A204135 p(2): .58, 3.41
%e A204135 p(3): .36, 1.44, 15.19
%e A204135 p(4): .21, .87, 4.53, 74.3
%e A204135 p(5): .12, .59, 2.14, 17.22, 380.91
%t A204135 f[i_, 1] := 1; f[1, j_] := 1;
%t A204135 f[i_, j_] := f[i, j - 1] + f[i - 1, j - 1] + f[i - 1, j]
%t A204135 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204135 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204135 Flatten[Table[f[i, n + 1 - i],
%t A204135   {n, 1, 15}, {i, 1, n}]]  (*  Delannoy, A008288 *)
%t A204135 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204135 c[n_] := CoefficientList[p[n], x]
%t A204135 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204135 Table[c[n], {n, 1, 8}]
%t A204135 Flatten[%]                 (* 204135 *)
%t A204135 TableForm[Table[c[n], {n, 1, 6}]]
%Y A204135 Cf. A008288, A202605, A204016.
%K A204135 tabl,sign
%O A204135 1,3
%A A204135 _Clark Kimberling_, Jan 12 2012