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

%I A136163 #3 Mar 30 2012 17:34:22
%S A136163 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,
%T A136163 -1,-35,0,210,0,-336,0,160,8,0,-168,0,672,0,-896,0,384,-1,63,0,-672,0,
%U A136163 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
%N A136163 Integration of A053120: triangle of coefficients of integration of Chebyshev's T(n,x) polynomials (powers of x in increasing order).
%C A136163 The row sums are:
%C A136163 {-2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2}
%C A136163 These polynomials are orthogonal:
%C A136163 Table[Table[Integrate[Sqrt[1/(1 - x^2)]*a0[[ n]]*a0[[m]], {x, -1, 1}], {n, 1, 11}], {m, 1, 11}]
%C A136163 Solving for the recurrence:
%C A136163 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}];
%C A136163 gives:
%C A136163 Q(x,n)=2*x*Q(x,n-1)-Q(x,n-2)
%D A136163 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.
%D A136163 Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 8 and pages 42 - 43;
%F A136163 T(x,n)=2*x*T(x,n-1)-T(x,n-2); Q(x,n)=Integrate[T(y,n-1),{y,-1,x}]
%e A136163 {1},
%e A136163 {-1, -1},
%e A136163 {},
%e A136163 {-1, -3, 0, 2},
%e A136163 {4,0, -12, 0, 8},
%e A136163 {-1, 15, 0, -40, 0, 24},
%e A136163 {-4, 0, 60, 0, -120, 0, 64},
%e A136163 {-1, -35, 0, 210, 0, -336, 0, 160},
%e A136163 {8, 0, -168, 0,672, 0, -896, 0, 384},
%e A136163 {-1, 63, 0, -672, 0, 2016, 0, -2304, 0, 896}.
%e A136163 {-8, 0, 360, 0, -2400, 0, 5760, 0, -5760, 0, 2048},
%e A136163 {-1, -99, 0, 1650, 0, -7920, 0, 15840, 0, -14080, 0, 4608}
%t A136163 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]
%Y A136163 Cf. A053120.
%K A136163 uned,tabf,sign
%O A136163 1,5
%A A136163 _Roger L. Bagula_, Mar 16 2008, corrected Apr 06 2008