1,5

The row sums are:

{-2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2}

These polynomials are orthogonal:

Table[Table[Integrate[Sqrt[1/(1 - x^2)]*a0[[ n]]*a0[[m]], {x, -1, 1}], {n, 1, 11}], {m, 1, 11}]

Solving for the recurrence:

Table[{c, d} /. Solve[{a0[[n]] -c*x*a0[[n - 1]] + d*a0[[n - 2]] == 0, a0[[n + 1]] - c*x*a0[[n]] + d*a0[[n - 1]] == 0}, {c, d}], {n, 3, 8}];

gives:

Q(x,n)=2*x*Q(x,n-1)-Q(x,n-2)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 8 and pages 42 - 43;

T(x,n)=2*x*T(x,n-1)-T(x,n-2); Q(x,n)=Integrate[T(y,n-1),{y,-1,x}]

{1},

{-1, -1},

{},

{-1, -3, 0, 2},

{4,0, -12, 0, 8},

{-1, 15, 0, -40, 0, 24},

{-4, 0, 60, 0, -120, 0, 64},

{-1, -35, 0, 210, 0, -336, 0, 160},

{8, 0, -168, 0,672, 0, -896, 0, 384},

{-1, 63, 0, -672, 0, 2016, 0, -2304, 0, 896}.

{-8, 0, 360, 0, -2400, 0, 5760, 0, -5760, 0, 2048},

{-1, -99, 0, 1650, 0, -7920, 0, 15840, 0, -14080, 0, 4608}

P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; a0 = Table[ExpandAll[P[x, n]] /. x -> y, {n, 0, 10}]; b0 = Table[n*(n - 2)*Integrate[a0[[n]], {y, -1, x}], {n, 1, 11}] a = Join[{{1}}, Table[CoefficientList[b0[[n]], x], {n, 1, 11}]] Table[Apply[Plus, CoefficientList[b0[[n]], x]], {n, 1, 11}] Flatten[a]

Cf. A053120.

Sequence in context: A159977 A112974 A113069 this_sequence A058624 A145856 A092154

Adjacent sequences: A136160 A136161 A136162 this_sequence A136164 A136165 A136166

uned,tabf,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 16 2008, corrected Apr 06 2008