A136330 A triangular sequence from 2^n times the coefficients of characteristic polynomials of a rational tridiagonal matrix type: M(3)= {{1/2,-1,0} {-1,1/2,-m}, {0,-1,1/2}}};m=-1; polynomial recursion associated is: p(x, n) = (1 - 2*x)*p(x, n - 1)/2 - p(x, n - 2);.

1, 1, -2, -3, -4, 4, -7, 10, 12, -8, 5, 40, -24, -32, 16, 33, -10, -152, 48, 80, -32, 13, -236, -36, 480, -80, -192, 64, -119, -222, 1044, 360, -1360, 96, 448, -128, -171, 960, 1632, -3648, -1760, 3584, 0, -1024, 256, 305, 2190, -4464, -8352, 10976, 6720, -8960, -512, 2304, -512, 989, -2260, -15372, 15168, 34720

Offset

1,3

Comments

Row sums are (2^n times) : {1, -1, -3, 7, 5, -33, 13, 119, -171, -305, 989}.

This type of matrix was suggested to me by the Cartan matrix type with the factor (2-x) in the recursion which has rational opposite (1-2*x).

Links

Table of n, a(n) for n=1..60.

Formula

Matrix: M(n,m,d)=If[ n == m, 1/2, If[n == d && m ==d - 1, -1, If[(n == m - 1 || n == m + 1), -1, 0]]]; out_n,m=Coefficient(2^n*CharacteristicPolynomial[M(n,m,d)); Polynomial recursion: p(x, n) = (1 - 2*x)*p(x, n - 1)/2 - p(x, n - 2); out_n,m=Coefficient(2^n*p(x,n)).

Example

{1},
{1, -2},
{-3, -4, 4},
{-7, 10, 12, -8},
{5, 40, -24, -32, 16},
{33, -10, -152, 48, 80, -32},
{13, -236, -36, 480, -80, -192, 64},
{-119, -222, 1044, 360, -1360, 96, 448, -128},
{-171, 960, 1632, -3648, -1760, 3584, 0, -1024, 256},
{305, 2190, -4464, -8352, 10976, 6720, -8960, -512, 2304, -512},
{989, -2260, -15372, 15168, 34720, -29568, -22400, 21504, 2304, -5120, 1024}

Mathematica

T[n_, m_, d_] := If[ n == m, 1/2, If[n == d && m == d - 1, -1, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a1 = Table[M[d], {d, 1, 10}]; Table[2^d*Det[M[d]], {d, 1, 10}]; g = Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[2^d*CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] (* Polynomial recursion*) Clear[p] p[x, 0] = 1; p[x, 1] = (1 - 2*x)/2; p[x, 2] = (-3 - 4 x + 4*x^2)/4; p[x_, n_] := p[x, n] = (1 - 2*x)*p[x, n - 1]/2 - p[x, n - 2]; Table[ExpandAll[2^n*p[x, n]], {n, 0, Length[g] - 1}]; Flatten[Table[CoefficientList[2^n*p[x, n], x], {n, 0, Length[g] - 1}]]

Crossrefs

Sequence in context: A006087 A364675 A241315 * A294267 A301763 A236129

Adjacent sequences: A136327 A136328 A136329 * A136331 A136332 A136333

Keyword

tabl,uned,sign

Author

Roger L. Bagula, Apr 12 2008

Status

approved