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 1-7 of 7 results.

A296464 Expansion of e.g.f. arcsin(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, 2, 28, 1024, 71632, 8192736, 1392793920, 330041217024, 104069101383936, 42159457593506304, 21346870862961183744, 13213529766600134344704, 9818417126704155249954816, 8625630408510010165396070400, 8844234850947343105068735283200, 10467364426053362392901751845683200
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arcsin(arcsin(x)) = x/1! + 2*x^3/3! + 28*x^5/5! + 1024*x^7/7! + 71632*x^9/9! + ...

Crossrefs

Cf. A001818, A003712, A012063, A296466.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcSin[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 Log[Log[I x + Sqrt[1 - x^2]] + Sqrt[1 + 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.: arcsinh(arcsinh(x)) (odd powers only, absolute values).
E.g.f.: -i*log(log(i*x + sqrt(1 - x^2)) + sqrt(1 + log(i*x + sqrt(1 - x^2))^2)), where i is the imaginary unit (odd powers only).
a(n) ~ sqrt(2) * (2*n)! / (sqrt(Pi*sin(2)*n) * sin(1)^(2*n)). - Vaclav Kotesovec, Dec 13 2017

A296679 Expansion of e.g.f. arcsinh(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 1, 13, 341, 18649, 1599849, 205524837, 36391450941, 8546308276401, 2564025898856913, 957697868873929149, 435619128300038521893, 237104370189582892175241, 152148421079949399306125625, 113672892845152570858515803925, 97820056722556900357454981990925
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Examples

			arcsinh(arctanh(x)) = x/1! + x^3/3! + 13*x^5/5! + 341*x^7/7! + 18649*x^9/9! + 1599849*x^11/11! + ...

Crossrefs

Cf. A001818, A003706, A010050, A012254, A096717, A096718, A296464, A296466, A296677.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcSinh[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 16; Table[(CoefficientList[Series[Log[(Log[1 + x] - Log[1 - x])/2 + Sqrt[1 + (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsin(arctan(x)) (odd powers only, absolute values).
E.g.f.: log((log(1 + x) - log(1 - x))/2 + sqrt(1 + (log(1 + x) - log(1 - x))^2/4)) (odd powers only).

A296680 Expansion of e.g.f. arcsin(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 3, 53, 2359, 198953, 27412011, 5625656541, 1613676694239, 617477049181521, 304167421243513683, 187546541676182230149, 141512355477854459198343, 128265950128144233675269241, 137512081213377707268891639675, 172108297920263623816775456321325
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Examples

			arcsin(arctanh(x)) = x/1! + 3*x^3/3! + 53*x^5/5! + 2359*x^7/7! + 198953*x^9/9! + 27412011*x^11/11! + ...

Links

Crossrefs

Cf. A001818, A003717, A010050, A012134, A296464, A296466, A296679.

Programs

  • Maple
    S:= series(arcsin(arctanh(x)),x,52):
seq(coeff(S,x,n)*n!,n=1..51,2); # Robert Israel, Dec 18 2017
  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcSin[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 15; Table[(CoefficientList[Series[-I Log[(I/2) (Log[1 + x] - Log[1 - x]) + Sqrt[1 - (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsinh(arctan(x)) (odd powers only, absolute values).
E.g.f.: -i*log((i/2)*(log(1 + x) - log(1 - x)) + sqrt(1 - (log(1 + x) - log(1 - x))^2/4)), where i is the imaginary unit (odd powers only).

A296728 Expansion of e.g.f. arcsin(x*cos(x)) (odd powers only).

Original entry on oeis.org

1, -2, -16, 8, 12672, 571264, -44351360, -12355211520, -452681248768, 478190483394560, 132554796040912896, -18854516962334277632, -27186884683859043123200, -5502410397289951851773952, 6273206188133923322747420672, 5389680791235134726930445369344
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arcsin(x*cos(x)) = x/1! - 2*x^3/3! - 16*x^5/5! + 8*x^7/7! + 12672*x^9/9! + ...

Crossrefs

Cf. A001818, A009015, A009016, A009446, A009447, A009633, A009634, A012495, A012780, A101928, A191512, A296464, A296466, A296679, A296680, A296729, A296730, A296731, A296740.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcSin[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(asin(x*cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

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

A296729 Expansion of e.g.f. arcsin(x*cosh(x)) (odd powers only).

Original entry on oeis.org

1, 4, 44, 1912, 156816, 21506816, 4420845376, 1271132964480, 487161448339712, 239980527068474368, 147742478026391141376, 111153314734461183924224, 100339775128577885016985600, 107037870347952811373977239552, 133204585741561810426003651444736
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arcsin(x*cosh(x)) = x/1! + 4*x^3/3! + 44*x^5/5! + 1912*x^7/7! + 156816*x^9/9! + ...

Crossrefs

Cf. A001818, A009015, A009016, A009446, A009447, A009633, A009634, A012495, A012780, A101928, A191512, A296464, A296466, A296679, A296680, A296728, A296730, A296731, A296740.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcSin[x Cosh[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(asin(x*cosh(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x*cosh(x)).

A296741 Expansion of e.g.f. arcsin(x*sec(x)) (odd powers only).

Original entry on oeis.org

1, 4, 64, 2752, 237312, 34390016, 7512117248, 2302977392640, 942529341030400, 496287845973753856, 326775812392982937600, 263039306566659448242176, 254121613033387345942937600, 290175686081926976733941071872, 386599796043915196967089006968832
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arcsin(x*sec(x)) = x/1! + 4*x^3/3! + 64*x^5/5! + 2752*x^7/7! + 237312*x^9/9! + ...

Crossrefs

Cf. A001818, A003700, A009118, A009119, A009562, A009563, A009765, A009843, A102072, A191003, A296464, A296466, A296679, A296680, A296742, A296743.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcSin[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(asin(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

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

A296742 Expansion of e.g.f. arcsinh(x*sec(x)) (odd powers only).

Original entry on oeis.org

1, 2, 4, -8, 2448, 130976, -2342848, -239130240, 99052990720, 8918588764672, -2795242017684480, -92786315822417920, 279479081010906828800, -57316070780459900928, -39411396653183724314673152, 5932051008707372732672475136, 10689040617354387626585873252352
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arcsinh(x*sec(x)) = x/1! + 2*x^3/3! + 4*x^5/5! - 8*x^7/7! + 2448*x^9/9! + ...

Crossrefs

Cf. A001818, A003700, A009118, A009119, A009562, A009563, A009765, A009843, A102072, A191003, A296464, A296466, A296679, A296680, A296741, A296743.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[ArcSinh[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(asinh(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arcsinh(x*sec(x)).
Showing 1-7 of 7 results.

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