%I #18 Sep 17 2024 08:24:06

%S -1,-1,1,0,4,-6,-4,11,-6,1,6,-41,75,-60,74,-119,57,64,-93,47,-11,1,

%T 144,-492,886,-1076,489,618,-1063,1154,-1672,1618,-410,-682,785,-392,

%U 108,-16,1,744,-2567,3782,-1075,-6736,18095,-29241,29006,-12952,-3601,11554,-18942,24467,-17741,2907,6473,-6678,3357,-1026

%N A triangular sequence from a Beraha type recursive polynomial using 5 X 5 centered tridiagonal matrices with chromatic polynomial central roots to its characteristic polynomial.

%H S. Beraha, J. Kahane and N. J. Weiss, <a href="https://doi.org/10.1073/pnas.72.11.4209">Limits of zeros of recursively defined polynomials</a>, Proc Natl Acad Sci U S A. 1975 November; 72(11): 4209.

%e Triangle begins:

%e {-1},

%e {-1, 1},

%e {0, 4, -6, -4, 11, -6, 1},

%e {6, -41,75, -60, 74, -119, 57, 64, -93, 47, -11, 1},

%e {144, -492, 886, -1076, 489, 618, -1063, 1154, -1672, 1618, -410, -682, 785, -392, 108, -16, 1}

%t M[n_] := {{n, -1, 0, 0, 0}, {-1, n, -1, 0, 0}, {0, -1, n, -1, 0}, {0, 0, -1, n, -1}, {0, 0, 0, -1, n}};

%t q[n_, x_] = Factor[CharacteristicPolynomial[M[n], x]];

%t p[0, x] = -1; p[1, x] = x - 1; p[k_, x_] := p[k, x] = -Sum[q[n, x]*p[k - n, x], {n, 1, k - 1}]; Table[Expand[p[n, x]], {n, 0, 10}]

%t w = Table[CoefficientList[p[n, x], x], {n, 0, 10}];

%t Flatten[w]

%K uned,tabf,sign

%O 1,5

%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 29 2006