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A136265 Integral form of A053120 :Triangle of coefficients of Integral form Chebyshev's T(n, x) polynomials (powers of x in increasing order); Much improved version by use of the integro-differential recursive form over a previous attempt.

%I A136265 #3 Mar 30 2012 17:34:22
%S A136265 1,-1,2,0,-4,2,3,-2,-12,4,0,16,-6,-32,8,-5,2,60,-16,-80,16,0,-36,10,
%T A136265 192,-40,-192,32,7,-2,-168,36,560,-96,-448,64,0,64,-14,-640,112,1536,
%U A136265 -224,-1024,128,-9,2,360,-64,-2160,320,4032,-512,-2304,256,0,-100,18,1600,-240,-6720,864,10240,-1152,-5120,512,11,-2
%N A136265 Integral form of A053120 :Triangle of coefficients of Integral form Chebyshev's T(n, x) polynomials (powers of x in increasing order); Much improved version by use of the integro-differential recursive form over a previous attempt.
%C A136265 Row sums are:
%C A136265 Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n,
%C A136265 0, 10}]];
%C A136265 {1, 1, -2, -7, -14, -23, -34, -47, -62, -79, -98, -119}
%C A136265 Integration of the doubled functions is not orthogonal:
%C A136265 Table[Table[Integrate[Sqrt[1/(1 - x^2)]*(2*x*P[x, n] - Q[x, n])*(2*x*P[x, m] -
%C A136265 Q[x, m]), {x, -1, 1}], {n, 0, 10}], {m, 0, 10}]
%D A136265 Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986,Pages 42-50
%F A136265 P(x, n) = 2*x*P(x, n - 1) - P(x, n - 2); Q(x, n) := D[P[x, n + 1], x]=dp(x,n)/dx Output Integral form: IP(x,n)=2*x*p(x,n)-Q(x,n)
%e A136265 {1},
%e A136265 {-1, 2},
%e A136265 {0, -4, 2},
%e A136265 {3, -2, -12, 4},
%e A136265 {0, 16, -6, -32, 8},
%e A136265 {-5,2, 60, -16, -80, 16},
%e A136265 {0, -36, 10, 192, -40, -192, 32},
%e A136265 {7, -2, -168, 36, 560, -96, -448, 64},
%e A136265 {0, 64, -14, -640, 112, 1536, -224, -1024, 128},
%e A136265 {-9, 2, 360, -64, -2160, 320, 4032, -512, -2304, 256},
%e A136265 {0, -100, 18, 1600, -240, -6720, 864, 10240, -1152, -5120, 512},
%e A136265 {11, -2, -660, 100,6160, -800, -19712, 2240, 25344, -2560, -11264, 1024}
%t A136265 P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Q[x_, n_] := D[P[x, n + 1], x]; Table[ExpandAll[2*x*P[x, n] - Q[x, n]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[2*x*P[x, n] - Q[x, n], x], {n, 0, 10}]]; Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n, 0, 10}]]; Flatten[a]
%Y A136265 Cf. A053120.
%K A136265 uned,tabl,sign
%O A136265 1,3
%A A136265 _Roger L. Bagula_, Mar 18 2008