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 1-6 of 6 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

A296436 Expansion of e.g.f. log(1 + arcsin(x))*exp(x).

Original entry on oeis.org

0, 1, 1, 3, 0, 28, -85, 1029, -6440, 79136, -724305, 9982005, -118974856, 1858582100, -27126378357, 478338929509, -8227405849840, 162502213354272, -3209170996757057, 70409595412300877, -1566861832498793248, 37885426233247176772, -936732798302547171509, 24780850678372964078189
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 12 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + x^2/2! + 3*x^3/3! + 28*x^5/5! - 85*x^6/6! + 1029*x^7/7! - 6440*x^8/8! + ...

Links

Crossrefs

Cf. A009334, A009337, A009353, A189815, A291482, A296437.

Programs

  • Maple
    a:=series(log(1+arcsin(x))*exp(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + ArcSin[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(ox=O(x^30)); Vecrev(Pol(serlaplace(log(1 + asin(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017

Formula

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

A296979 Expansion of e.g.f. arcsin(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 3, -12, 68, -480, 4144, -42112, 494360, -6581880, 98079696, -1617373296, 29245459176, -575367843960, 12235339942344, -279650131845120, 6836254328079936, -177979145883651648, 4916243253642325056, -143602294106947553280, 4422411460743707222784
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arcsin(log(1 + x)) = x^1/1! - x^2/2! + 3*x^3/3! - 12*x^4/4! + 68*x^5/5! - 480*x^6/6! + ...

Crossrefs

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A189815, A296980, A296981, A296982.

Programs

  • Maple
    a:=series(arcsin(log(1+x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[ArcSin[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I Log[1 + x] + Sqrt[1 - Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ -(-1)^n * n^(n-1) / (exp(1) - 1)^(n - 1/2). - Vaclav Kotesovec, Mar 26 2019

A297209 Expansion of e.g.f. log(1 + arcsin(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 9, -32, 148, -853, 6027, -49576, 470624, -5005137, 59454923, -774282632, 11035740844, -169997137269, 2826070412955, -50256453936368, 954657085889760, -19247168446169665, 411277539407862707, -9269937746437524256, 220085825544691181500, -5483977295221312280757
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			log(1 + arcsin(x))*exp(-x) = x/1! - 3*x^2/2! + 9*x^3/3! - 32*x^4/4! + 148*x^5/5! - 853*x^6/6! + ...

Crossrefs

Cf. A009337, A009356, A009374, A009393, A189815, A296436, A297210, A297211, A297213.

Programs

  • Maple
    a:=series(log(1+arcsin(x))*exp(-x),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Log[1 + ArcSin[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    x='x+O('x^99); concat([0], Vec(serlaplace(exp(-x)*log(1+asin(x))))) \\ Altug Alkan, Dec 28 2017

A331615 E.g.f.: exp(1 / (1 - arcsin(x)) - 1).

Original entry on oeis.org

1, 1, 3, 14, 85, 640, 5703, 58760, 685353, 8925632, 128231627, 2014061568, 34312150525, 630043097216, 12400033125647, 260357810321664, 5807790344591953, 137144754146230272, 3417248676737769619, 89590823377278496768, 2465026658283881339301
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Crossrefs

Cf. A006228, A189780, A189815, A331607, A331608, A331616, A331617, A331618.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[1/(1 - ArcSin[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A189780[0] = 1; A189780[n_] := A189780[n] = Sum[Binomial[n, k] If[OddQ[k], ((k - 2)!!)^2, 0] A189780[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A189780[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)={Vec(serlaplace(exp(1/(1 - asin(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) * A189780(k) * a(n-k).

A296622 Expansion of e.g.f. log(1 + arcsin(x)*arcsinh(x)) (even powers only).

Original entry on oeis.org

0, 2, -12, 328, -15008, 1356192, -166628352, 31500831360, -7474571071488, 2418220114014720, -940432709166170112, 464609611973533501440, -268355615175956213268480, 188067307050238642631516160, -151072053399934628129585233920, 142618740583722182161589570273280
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 17 2017

Keywords

Examples

			log(1 + arcsin(x)*arcsinh(x)) = 2*x^2/2! - 12*x^4/4! + 328*x^6/6! - 15008*x^8/8! + 1356192*x^10/10! - 166628352*x^12/12! + ...

Crossrefs

Cf. A001818, A009232, A009359, A012598, A189815, A296435.

Programs

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

Formula

E.g.f.: log(1 - i*log(i*x + sqrt(1 - x^2))*log(x + sqrt(1 + x^2))), where i is the imaginary unit (even powers only).
Showing 1-6 of 6 results.

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