A123969 A triangular sequence from a Beraha type recursive polynomial using 5 X 5 n centered tridiagonal matrices with chromatic polynomial central roots to its characteristic polynomial.

-1, -1, 1, 0, 4, -6, -4, 11, -6, 1, 6, -41, 75, -60, 74, -119, 57, 64, -93, 47, -11, 1, 144, -492, 886, -1076, 489, 618, -1063, 1154, -1672, 1618, -410, -682, 785, -392, 108, -16, 1, 744, -2567, 3782, -1075, -6736, 18095, -29241, 29006, -12952, -3601, 11554, -18942, 24467, -17741, 2907, 6473, -6678, 3357, -1026

Offset

1,5

Comments

Recursive polynomial type from a classic paper related to Tutte-Beraha constants and the four color graph/ topology problem.

References

Limits of zeros of recursively defined polynomials, S. Beraha, J. Kahane and N. J. Weiss, Proc Natl Acad Sci U S A. 1975 November; 72(11): 4209.

Links

Table of n, a(n) for n=1..58.

Formula

M(n)={{n, -1, 0, 0, 0}, {-1, n, -1, 0, 0}, {0, -1, n, -1, 0}, {0, 0, -1, n, -1}, q(n,x)=CharacteristicPolynomial(M(n),x) p(k, x) = -Sum[q(n, x)*p(k - n, x), {n, 1, k - 1}]

Example

Triangle begins:
{-1},
{-1, 1},
{0, 4, -6, -4, 11, -6, 1},
{6, -41,75, -60, 74, -119, 57, 64, -93, 47, -11, 1},
{144, -492, 886, -1076, 489, 618, -1063, 1154, -1672, 1618, -410, -682, 785, -392, 108, -16, 1}

Mathematica

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}}; q[n_, x_] = Factor[CharacteristicPolynomial[M[n], x]]; 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}] w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

Crossrefs

Sequence in context: A328722 A143521 A278363 * A255679 A019188 A019244

Adjacent sequences: A123966 A123967 A123968 * A123970 A123971 A123972

Keyword

uned,tabf,sign

Author

Gary W. Adamson and Roger L. Bagula, Oct 29 2006

Status

approved