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Search: id:A136160
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| A136160 |
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Derivative of A053120 Triangle of coefficients of the derivative Chebyshev's T(n,x) polynomials (powers of x in increasing order). |
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| 1, 0, 4, -3, 0, 12, 0, -16, 0, 32, 5, 0, -60, 0, 80, 0, 36, 0, -192, 0, 192, -7, 0, 168, 0, -560, 0, 448, 0, -64, 0, 640, 0, -1536, 0, 1024, 9, 0, -360, 0, 2160, 0, -4032, 0, 2304, 0, 100, 0, -1600, 0, 6720, 0, -10240, 0, 5120, -11, 0, 660, 0, -6160, 0, 19712, 0, -25344, 0, 11264
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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The clue to relating the Steinbach golden field polynomials and the Boubaker
polynomials is in the Steinbach paper as reference.
This sequence of polynomials, the Boubaker polynomials and the Steinbach polynomials when integrated give the same alternating orthogonality.
Row sums are the squares:
{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121}
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REFERENCES
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Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 8 and pages 42 - 43
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.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
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FORMULA
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T(x, n) = 2*x*T(x, n - 1) - T(x, n - 2) Q(x,n)=dT(x,n+1)/dt
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EXAMPLE
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{1},
{0, 4},
{-3, 0, 12},
{0, -16, 0, 32},
{5, 0, -60, 0, 80},
{0, 36, 0, -192, 0, 192},
{-7, 0, 168, 0, -560, 0, 448},
{0, -64, 0, 640, 0, -1536,0, 1024},
{9, 0, -360, 0, 2160,0, -4032, 0, 2304},
{0, 100, 0, -1600, 0, 6720, 0, -10240, 0, 5120},
{-11, 0, 660, 0, -6160, 0, 19712, 0, -25344, 0, 11264}
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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]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Cf. A053120, A135929.
Adjacent sequences: A136157 A136158 A136159 this_sequence A136161 A136162 A136163
Sequence in context: A019756 A010650 A011091 this_sequence A120362 A010102 A054669
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 16 2008
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