cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A296467 Expansion of e.g.f. arctan(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 0, 8, 112, 8192, 599808, 80010240, 13537247232, 3160676007936, 929451393220608, 343173318976733184, 154043745649772986368, 82935056810462020632576, 52660879605487383997317120, 38970318170642827020431523840, 33236188662933234332228627988480, 32365907321554306913981616441262080
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arctan(arctanh(x)) =  x/1! + 8*x^5/5! + 112*x^7/7! + 8192*x^9/9! + 599808*x^11/11! + 80010240*x^13/13! + ...

Links

Crossrefs

Cf. A003721, A010050, A012231, A296465.

Programs

  • Maple
    S:= series(arctan(arctanh(x)),x,52):
seq(coeff(S,x,2*i+1)*(2*i+1)!,i=0..25); # Robert Israel, Dec 13 2017
  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[ArcTan[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 17; Table[(CoefficientList[Series[I (Log[2 + I Log[1 - x] - I Log[1 + x]] - Log[2 - I Log[1 - x] + I Log[1 + x]])/2, {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arctanh(arctan(x)) (odd powers only, absolute values).
E.g.f.: i*(log(2 + i*log(1 - x) - i*log(1 + x)) - log(2 - i*log(1 - x) + i*log(1 + x)))/2, where i is the imaginary unit (odd powers only).

A296730 Expansion of e.g.f. arctanh(x*cos(x)) (odd powers only).

Original entry on oeis.org

1, -1, -31, -337, 24705, 2451679, -17936543, -42895630065, -5396647099903, 1239561882325439, 708575518706816481, 37448619025871342959, -113842057082636742446975, -52054011876398495316250977, 16226448322449614832534708065, 31975745831751940004484917311439
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arctanh(x*cos(x)) = x/1! - x^3/3! - 31*x^5/5! - 337*x^7/7! + 24705*x^9/9! + ...

Crossrefs

Cf. A000364, A009015, A009016, A009446, A009447, A009633, A009634, A010050, A012494, A191512, A296465, A296467, A296728, A296729, A296731, A296740.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcTanh[x Cos[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(atanh(x*cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arctanh(x*cos(x)).

A296743 Expansion of e.g.f. arctanh(x*sec(x)) (odd powers only).

Original entry on oeis.org

1, 5, 109, 5977, 612729, 100954061, 24395453861, 8128143367905, 3571195811862385, 2000535014776893973, 1391684597704875555165, 1177047158822263838854889, 1189444022487013498606939625, 1415364934488337503351305867997, 1958850511524588636608881908473749
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arctanh(x*sec(x)) = x/1! + 5*x^3/3! + 109*x^5/5! + 5977*x^7/7! + 612729*x^9/9! + ...

Crossrefs

Cf. A003700, A009118, A009119, A009562, A009563, A009765, A009843, A010050, A102075, A191003, A296465, A296467, A296741, A296742.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcTanh[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(atanh(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arctanh(x*sec(x)).

A296677 Expansion of e.g.f. arctan(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, -1, 13, -173, 12409, -370137, 88556037, -2668274373, 2491377242481, 34526890553679, 202383113207336829, 25792743610973373219, 39172126704113226631401, 12501799823936578879327095, 15717805122762984314778029685, 9078237580992214462785729689355
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Examples

			arctan(arcsin(x)) = x/1! - x^3/3! + 13*x^5/5! - 173*x^7/7! + 12409*x^9/9! - 370137*x^11/11! + ...

Crossrefs

Cf. A001818, A003705, A010050, A012194, A096719, A096720, A296465, A296467.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcTan[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 16; Table[(CoefficientList[Series[(I/2) Log[1 - Log[I x + Sqrt[1 - x^2]]] - (I/2) Log[1 + Log[I x + Sqrt[1 - x^2]]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: (i/2)*log(1 - log(i*x + sqrt(1 - x^2))) - (i/2)*log(1 + log(i*x + sqrt(1 - x^2))), where i is the imaginary unit (odd powers only).

A296678 Expansion of e.g.f. arctanh(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, 3, 53, 2303, 185033, 23756667, 4457821821, 1150764459063, 391167511473681, 169370797497060339, 91013260219635394629, 59435772666287730632559, 46362471059282707504957401, 42577231265939498962852834155, 45471686987452309473064526678925
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Examples

			arctanh(arcsin(x)) = x/1! + 3*x^3/3! + 53*x^5/5! + 2303*x^7/7! + 185033*x^9/9! + 23756667*x^11/11! + ...

Crossrefs

Cf. A001818, A003716, A010050, A012258, A296465, A296467, A296677.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcTanh[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 15; Table[(CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]]/2 - Log[1 + I Log[I x + Sqrt[1 - x^2]]]/2, {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arctan(arcsinh(x)) (odd powers only, absolute values).
E.g.f.: log(1 - i*log(i*x + sqrt(1 - x^2)))/2 - log(1 + i*log(i*x + sqrt(1 - x^2)))/2, where i is the imaginary unit (odd powers only).
Showing 1-5 of 5 results.

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