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.

A296435 Expansion of e.g.f. log(1 + arcsinh(x)).

Original entry on oeis.org

0, 1, -1, 1, -2, 13, -64, 173, -720, 12409, -114816, 370137, -1491456, 88556037, -1263184896, 2668274373, 21448022016, 2491377242481, -50233550831616, -34526890553679, 5153298175033344, 202383113207336829, -5453228045913292800, -25792743610973373219, 1393299559788718325760
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 12 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! - x^2/2! + x^3/3! - 2*x^4/4! + 13*x^5/5! - 64*x^6/6! + ...

Links

Crossrefs

Cf. A001818, A003704, A110708, A189815, A202139, A296437.

Programs

  • Maple
    S:= series(ln(1+arcsinh(x)),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Dec 12 2017
  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 + ArcSinh[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Log[1 + Log[x + Sqrt[1 + x^2]]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    Vecrev(Pol(serlaplace(log(1 + asinh(x + O(x^30)))))) \\ Andrew Howroyd, Dec 12 2017

Formula

E.g.f.: log(1 + log(x + sqrt(1 + x^2))).
a(n) ~ 4*(Pi*cos(Pi*n/2) + 2*sin(Pi*n/2)) * n^(n-1) / ((4 + Pi^2) * exp(n)). - Vaclav Kotesovec, Dec 21 2017

A296438 Expansion of e.g.f. log(1 + arctan(x))*exp(x).

Original entry on oeis.org

0, 1, 1, 0, 0, 13, 5, -336, -56, 18593, -6735, -1598520, 1192664, 205475645, -255011835, -36324220856, 62049925040, 8519764352097, -18835422533375, -2551646722754512, 6927586371061712, 951619735931190157, -3077560879933239899, -432185107142832520576, 1624964470900980885432
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 12 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + x^2/2! + 13*x^5/5! + 5*x^6/6! - 336*x^7/7! - 56*x^8/8! + ...

Links

Crossrefs

Cf. A009371, A009374, A009390, A110708, A279927, A296439.

Programs

  • Maple
    a:=series(log(1+arctan(x))*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 + ArcTan[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Log[1 + I (Log[1 - I x] - Log[1 + I x])/2] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(ox=O(x^30)); Vecrev(Pol(serlaplace(log(1 + atan(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017

Formula

E.g.f.: log(1 + i*(log(1 - i*x) - log(1 + i*x))/2)*exp(x), where i is the imaginary unit.

A331617 E.g.f.: exp(1 / (1 - arctan(x)) - 1).

Original entry on oeis.org

1, 1, 3, 11, 49, 265, 1683, 12035, 95169, 832337, 7998467, 83033403, 922112305, 10978263257, 139956480467, 1889161216179, 26798589518593, 401123509624737, 6346168059440515, 105040097140558699, 1805102151607613361, 32421358229074354601
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Comments

a(53) is negative. - Vaclav Kotesovec, Jan 26 2020

Links

Crossrefs

Cf. A002019, A110708, A191700, A331610, A331615, A331616, A331618.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[1/(1 - ArcTan[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A191700[0] = 1; A191700[n_] := A191700[n] = Sum[Binomial[n, k] If[OddQ[k], (-1)^Boole[IntegerQ[(k + 1)/4]] (k - 1)!, 0] A191700[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A191700[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
  • PARI
    seq(n)={Vec(serlaplace(exp(1/(1 - atan(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A191700(k) * a(n-k).

A354117 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + arctan(x).

Original entry on oeis.org

1, -2, -2, 8, -16, 176, -832, 384, 8192, 447744, -4228608, -15860736, -398991360, 10938421248, 44581613568, -29064658944, -17762113880064, -18092698632192, -7331825098948608, -64037289416196096, 3154526750647517184, 91791873021766533120, -1278044473427380666368
Offset: 1

Views

Author

Ilya Gutkovskiy, May 17 2022

Keywords

Crossrefs

Cf. A010050, A110708, A353820, A353915, A354065, A354115, A354116, A354118.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + ArcTan[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

E.g.f.: Sum_{k>=1} mu(k) * log(1 + arctan(x^k)) / k.

A296981 Expansion of e.g.f. arctan(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 0, 6, -22, -30, 952, -5656, -9952, 508320, -3874992, -20690208, 833780400, -7697940432, -52230156288, 2467649024640, -24686997151104, -329724479772288, 14493628861307136, -159114034671287040, -2682505451050592256, 126421889770129637376
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arctan(log(1 + x)) = x^1/1! - x^2/2! + 6*x^4/4! - 22*x^5/5! - 30*x^6/6! + ...

Crossrefs

Cf. A001710, A003703, A003708, A009024, A009454, A009775, A010050, A104150, A110708, A296979, A296980, A296982.

Programs

  • Maple
    a:=series(arctan(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[ArcTan[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[(I/2) Log[1 - I Log[1 + x]] - (I/2) Log[1 + I Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ (-1)^(n+1) * (n-1)! * sin(n*(Pi-1)/2) / (2 - 2*cos(1))^(n/2). - Vaclav Kotesovec, Mar 26 2019

A297211 Expansion of e.g.f. log(1 + arctan(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 6, -8, 13, -103, 462, 824, -8239, -147747, 1233518, 12148288, -127674419, -2090702391, 24495009510, 410685350032, -5514147250815, -111860639828131, 1673006899192118, 37306857729115304, -619246417449233555, -15476404474443728487, 281907759055194714206
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			log(1 + arctan(x))*exp(-x) = x/1! - 3*x^2/2! + 6*x^3/3! - 8*x^4/4! + 13*x^5/5! - 103*x^6/6! + ...

Crossrefs

Cf. A009337, A009356, A009374, A009393, A110708, A296438, A297209, A297210, A297213.

Programs

  • Maple
    a:=series(log(1+arctan(x))*exp(-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + ArcTan[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + (I/2) (Log[1 - I x] - Log[1 + I x])] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

A296623 Expansion of e.g.f. log(1 + arctan(x)*arctanh(x)) (even powers only).

Original entry on oeis.org

0, 2, -12, 448, -21728, 2380032, -318185472, 69695846400, -18235768762368, 6697099792220160, -2892199532135841792, 1606188416621920911360, -1034069421398404544593920, 810882197441673837894696960, -727447103613537543910242385920, 766865924510666637669136261447680
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 17 2017

Keywords

Examples

			log(1 + arctan(x)*arctanh(x)) = 2*x^2/2! - 12*x^4/4! + 448*x^6/6! - 21728*x^8/8! + 2380032*x^10/10! - 318185472*x^12/12! + ...

Crossrefs

Cf. A009272, A009398, A010050, A012727, A110708, A202139.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Log[1 + ArcTan[x] ArcTanh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Log[1 + (I/4) (Log[1 - I x] - Log[1 + I x]) (Log[1 + x] - Log[1 - x])], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

E.g.f.: log(1 + (i/4)*(log(1 - i*x) - log(1 + i*x))*(log(1 + x) - log(1 - x))), where i is the imaginary unit (even powers only).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.