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Search: id:A137668
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| A137668 |
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Triangular sequence of coefficients from the expansion of p(x,t)=Tan[x*ArcTan[t]] which is in the Chebyshevlike form : T(t,x)=Cos(x*ArcCos(t)). |
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| 0, 1, 0, -2, 0, 2, 0, 24, 0, -40, 0, 16, 0, -720, 0, 1568, 0, -1120, 0, 272, 0, 40320, 0, -104704, 0, 102144, 0, -45696, 0, 7936, 0, -3628800, 0, 10720512, 0, -12869120, 0, 8042496, 0, -2618880, 0, 353792, 0, 479001600, 0, -1565051904, 0, 2188865536, 0, -1712668672, 0, 789854208, 0, -202369024, 0
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Row sums are: {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}.
The polynomials are not orthogonal on a Chebyshev weighted domain.
The Sederberg-Chang nested function: f^n(x)=Tan[2^n*ArcTan[x]] has the period doubling type of dynamics found in quadratics in complex dynamics: e.g. the Mandelbrot-Julia Pc polynomials.
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REFERENCES
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Chang and Sederberg, Over and Over Again, MAA, 1997, page 111
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FORMULA
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p(x,t)=Tan[x*ArcTan[t]]=Sum(P(x,n)*t^n/n!,{n,0,Infinity}]; out_n,m=n!*Coefficients(p(x,n)) odd terms only.
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EXAMPLE
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{0, 1},
{0, -2, 0, 2},
{0, 24, 0, -40, 0, 16},
{0, -720, 0, 1568, 0, -1120, 0, 272},
{0, 40320, 0, -104704, 0, 102144, 0, -45696, 0, 7936},
{0, -3628800, 0, 10720512, 0, -12869120, 0, 8042496, 0, -2618880, 0, 353792}, {0, 479001600, 0, -1565051904, 0, 2188865536, 0, -1712668672, 0, 789854208, 0, -202369024, 0, 22368256},
{0, -87178291200,0, 309188763648, 0, -487356047360, 0, 450481647616,0, -263012372480, 0, 96327655424, 0, -20355112960, 0, 1903757312},
{0, 20922789888000, 0, -79493016453120, 0, 138125290635264, 0, -145543597588480, 0, 101310804328448, 0, -47338162094080, 0, 14395135885312, 0, -2589109944320, 0, 209865342976},
{0, -6402373705728000, 0, 25804966598737920, 0,-48657759347146752, 0, 57064887390568448, 0, -45634645720694784, 0, 25589689363070976, 0, -9984529525374976, 0, 2597395096141824, 0, -406719034687488, 0, 29088885112832}
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MATHEMATICA
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Clear[p, b, a] p[t_] = Tan[x*ArcTan[t]]; g = Table[ ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 1, 20, 2}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 1, 20, 2}]; Flatten[a]
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CROSSREFS
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Sequence in context: A057498 A137949 A019214 this_sequence A056615 A060989 A135298
Adjacent sequences: A137665 A137666 A137667 this_sequence A137669 A137670 A137671
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KEYWORD
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tabf,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 27 2008
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