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Search: id:A136265
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| A136265 |
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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. |
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1
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| 1, -1, 2, 0, -4, 2, 3, -2, -12, 4, 0, 16, -6, -32, 8, -5, 2, 60, -16, -80, 16, 0, -36, 10, 192, -40, -192, 32, 7, -2, -168, 36, 560, -96, -448, 64, 0, 64, -14, -640, 112, 1536, -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
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row sums are:
Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n,
0, 10}]];
{1, 1, -2, -7, -14, -23, -34, -47, -62, -79, -98, -119}
Integration of the doubled functions is not orthogonal:
Table[Table[Integrate[Sqrt[1/(1 - x^2)]*(2*x*P[x, n] - Q[x, n])*(2*x*P[x, m] -
Q[x, m]), {x, -1, 1}], {n, 0, 10}], {m, 0, 10}]
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REFERENCES
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Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986,Pages 42-50
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FORMULA
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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)
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EXAMPLE
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{1},
{-1, 2},
{0, -4, 2},
{3, -2, -12, 4},
{0, 16, -6, -32, 8},
{-5,2, 60, -16, -80, 16},
{0, -36, 10, 192, -40, -192, 32},
{7, -2, -168, 36, 560, -96, -448, 64},
{0, 64, -14, -640, 112, 1536, -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, -660, 100,6160, -800, -19712, 2240, 25344, -2560, -11264, 1024}
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A053120.
Adjacent sequences: A136262 A136263 A136264 this_sequence A136266 A136267 A136268
Sequence in context: A037035 A112824 A001100 this_sequence A066910 A094405 A028609
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 18 2008
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