A137335 Triangular array read by rows, from polynomial recursion for every other term of Chebyshev orthogonal polynomials of the second kind: U(x,n)=Sin((n+1)*ArcSin(x))/Sin(ArcSin(x)) As q(x,n)=-2*(-1+2*x^2)*q(x,n-1)-q(x,n-1).

1, 3, 0, -4, 5, 0, -20, 0, 16, 7, 0, -56, 0, 112, 0, -64, 9, 0, -120, 0, 432, 0, -576, 0, 256, 11, 0, -220, 0, 1232, 0, -2816, 0, 2816, 0, -1024, 13, 0, -364, 0, 2912, 0, -9984, 0, 16640, 0, -13312, 0, 4096, 15, 0, -560, 0, 6048, 0, -28800, 0, 70400, 0, -92160, 0, 61440, 0, -16384, 17, 0, -816, 0, 11424, 0, -71808, 0

Offset

1,2

Comments

Alternative code:

Table[Normal[Series[Sin[(n + 1)*ArcSin[x]]/Sin[ArcSin[x]], {x, 0, 30}]], {n, 0, 10, 2}]

Equivalent to Sin[(2*n+1)*ArcSin[x]] recursion divided by x:

(* odd term Sin[n*ArcSin[x] *)

Clear[p]

p[x, 0] = x; p[x, 1] = 3*x - 4*x^3;

p[x_, n_] := p[x, n] = -2*(-1 + 2*x^2)*p[x, n - 1] - p[x, n - 2];

Table[ExpandAll[p[x, n]], {n, 0, 10}]

This odd term form integrates as orthogonal where this Rosenblum and Rovnyak alternating form doesn't.

Table[Integrate[q[x, n]*q[x, m]/Sqrt[1 - x^2], {x, -1, 1}], {n, 0, 10}, {m, 0, 10}]

Row sums are:

{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1}

References

Rosenblum and Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18-19

Links

Table of n, a(n) for n=1..72.

Formula

q(x, 0) = 1; q(x, 1) = 3 - 4*x^2; q(x,n)=-2*(-1+2*x^2)*q(x,n-1)-q(x,n-1) ( starting terms are important here: q(x, 0) = x; q(x, 1) = 3x - 4*x^3; gives a 'better' sequence)

Example

{1},
{3, 0, -4},
{5, 0, -20, 0, 16},
{7, 0, -56,0, 112, 0, -64},
{9, 0, -120, 0, 432, 0, -576, 0, 256},
{11, 0, -220,0, 1232, 0, -2816, 0, 2816, 0, -1024},
{13, 0, -364, 0, 2912,0, -9984, 0, 16640, 0, -13312, 0, 4096},
{15,0, -560, 0, 6048, 0, -28800, 0, 70400, 0, -92160, 0, 61440, 0, -16384},
{17, 0, -816, 0, 11424, 0, -71808, 0, 239360, 0, -452608, 0, 487424, 0, -278528, 0, 65536},
{19, 0, -1140, 0,20064, 0, -160512, 0, 695552, 0, -1770496, 0, 2723840, 0, -2490368, 0, 1245184, 0, -262144},
{21, 0, -1540, 0, 33264, 0, -329472,0, 1793792, 0, -5870592, 0, 12042240, 0, -15597568, 0, 12386304, 0, -5505024, 0, 1048576}

Mathematica

Clear[q] q[x, 0] = 1; q[x, 1] = 3 - 4*x^2; q[x_, n_] := q[x, n] = -2*(-1 + 2*x^2)*q[x, n - 1] - q[x, n - 2]; Table[ExpandAll[q[x, n]], {n, 0, 10}]; a = Table[CoefficientList[q[x, n], x], {n, 0, 10}] Flatten[a]

Crossrefs

Sequence in context: A354783 A209132 A019789 * A011077 A322936 A307383

Adjacent sequences: A137332 A137333 A137334 * A137336 A137337 A137338

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Apr 07 2008

Status

approved