1,2

Row sums are: {1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, ...}

They aren't orthogonal on a Chebyshev domain and weight function, but Gram-Schmidt says the vectors are orthogonal. Recursion polynomial analysis says that you need three starting polynomials for the recursion to work.

Theory would indicate that these are "like" B_n and A_n Cartan matrices and are probably orthogonal polynomials, but that doesn't give a domain or weight function.

Matrix definition: t(n,m,d)=If[ n == m, 2, If[n == d && m == d - 1, 0, If[(n == m - 1 || n == m + 1), -1, 0]]];

G.f.: g(x,t)=(2 - x)/(1 - (2 - x)*t + t^2).

{1},

{2, -1},

{4, -4, 1},

{6, -11, 6, -1},

{8, -24, 22, -8, 1},

{10, -45, 62, -37, 10, -1},

{12, -76, 147, -128, 56, -12, 1},

{14, -119,308, -366, 230, -79, 14, -1},

{16, -176, 588, -912, 770, -376, 106, -16, 1},

{18, -249, 1044, -2046, 2222, -1443, 574, -137, 18, -1},

{20, -340, 1749, -4224, 5720, -4732, 2485, -832, 172, -20, 1}

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

p[t_] = (2 - x)/(1 - (2 - x)*t + t^2); Table[ ExpandAll[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

Cf. A053122.

Sequence in context: A115237 A105542 A136600 this_sequence A097750 A133544 A013609

Adjacent sequences: A136669 A136670 A136671 this_sequence A136673 A136674 A136675

nonn,uned,tabl

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 04 2008, Apr 12 2008