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Search: id:A137307
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| A137307 |
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A triangular sequence of coefficients of even plus odd Chebyshev polynomials, A053120: q(x,n)=T(x,2*n-1]+T(x,2*n). |
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1
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| 1, 1, -1, 1, 2, 1, -3, -8, 4, 8, -1, 5, 18, -20, -48, 16, 32, 1, -7, -32, 56, 160, -112, -256, 64, 128, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512, 1, -11, -72, 220, 840, -1232, -3584, 2816, 6912, -2816, -6144, 1024, 2048, -1, 13, 98, -364, -1568, 2912, 9408, -9984, -26880, 16640, 39424, -13312, -28672
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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The row sums are all 2, and double integrations are all orthogonal except for the zero to one level.
This arose from an idea of Chladni Chebyshev's:
q(Exp[i*t],n)=T(Cos[2*Pi*t),2*n-1)+T(Sin(2*Pi*t),2*n)
which are strange looping spirals.
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FORMULA
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q(x,n)=T(x,2*n-1]+T(x,2*n).
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EXAMPLE
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{1, 1},
{-1, 1, 2},
{1, -3, -8, 4, 8},
{-1, 5,18, -20, -48, 16, 32},
{1, -7, -32, 56, 160, -112, -256, 64, 128},
{-1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512},
{1, -11, -72, 220, 840, -1232, -3584, 2816, 6912, -2816, -6144, 1024, 2048},
{-1, 13, 98, -364, -1568, 2912, 9408, -9984, -26880, 16640, 39424, -13312, -28672, 4096, 8192},
{1, -15, -128, 560, 2688, -6048, 21504, 28800, 84480, -70400, -180224, 92160, 212992, -61440, -131072, 16384, 32768},
{-1, 17, 162, -816, -4320, 11424, 44352, -71808, -228096, 239360, 658944, -452608, -1118208, 487424, 1105920, -278528, -589824, 65536, 131072},
{1, -19, -200, 1140, 6600, -20064, -84480, 160512, 549120, -695552, -2050048, 1770496, 4659200, -2723840, -6553600, 2490368, 5570560, -1245184, -2621440, 262144, 524288}
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MATHEMATICA
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Q[x_, n_] := ChebyshevT[2*n - 1, x] + ChebyshevT[2*n, x]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a0]
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CROSSREFS
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Cf. A053120.
Adjacent sequences: A137304 A137305 A137306 this_sequence A137308 A137309 A137310
Sequence in context: A135299 A092081 A057740 this_sequence A078045 A057300 A076655
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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), Apr 20 2008
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