A136163 Integration of A053120: triangle of coefficients of integration of Chebyshev's T(n,x) polynomials (powers of x in increasing order).

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

Offset

1,5

Comments

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)

References

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;

Links

Table of n, a(n) for n=1..75.

Formula

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

Example

{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}

Mathematica

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]

Crossrefs

Cf. A053120.

Sequence in context: A201924 A112974 A113069 * A178313 A190013 A171088

Adjacent sequences: A136160 A136161 A136162 * A136164 A136165 A136166

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Mar 16 2008, corrected Apr 06 2008

Status

approved