A137335 - OEIS

%I #3 Mar 30 2012 17:34:26

%S 1,3,0,-4,5,0,-20,0,16,7,0,-56,0,112,0,-64,9,0,-120,0,432,0,-576,0,

%T 256,11,0,-220,0,1232,0,-2816,0,2816,0,-1024,13,0,-364,0,2912,0,-9984,

%U 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

%N 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).

%C Alternative code:

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

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

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

%C Clear[p]

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

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

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

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

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

%C Row sums are:

%C {1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1}

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

%F 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)

%e {1},

%e {3, 0, -4},

%e {5, 0, -20, 0, 16},

%e {7, 0, -56,0, 112, 0, -64},

%e {9, 0, -120, 0, 432, 0, -576, 0, 256},

%e {11, 0, -220,0, 1232, 0, -2816, 0, 2816, 0, -1024},

%e {13, 0, -364, 0, 2912,0, -9984, 0, 16640, 0, -13312, 0, 4096},

%e {15,0, -560, 0, 6048, 0, -28800, 0, 70400, 0, -92160, 0, 61440, 0, -16384},

%e {17, 0, -816, 0, 11424, 0, -71808, 0, 239360, 0, -452608, 0, 487424, 0, -278528, 0, 65536},

%e {19, 0, -1140, 0,20064, 0, -160512, 0, 695552, 0, -1770496, 0, 2723840, 0, -2490368, 0, 1245184, 0, -262144},

%e {21, 0, -1540, 0, 33264, 0, -329472,0, 1793792, 0, -5870592, 0, 12042240, 0, -15597568, 0, 12386304, 0, -5505024, 0, 1048576}

%t 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]

%K uned,tabf,sign

%O 1,2

%A _Roger L. Bagula_, Apr 07 2008