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.

A296437 Expansion of e.g.f. log(1 + arcsinh(x))*exp(x).

Original entry on oeis.org

0, 1, 1, 1, 0, 8, -5, -51, -504, 8224, -12865, -296155, -2166736, 73348780, -116217309, -7440979651, -39733320080, 2564082122752, -3056854891489, -544155777899859, -2138400746459448, 251904027415707852, -163714875656114029, -92626483427571793931, -273784346863222483272
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 12 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + x^2/2! + x^3/3! + 8*x^5/5! - 5*x^6/6! - 51*x^7/7! - 504*x^8/8! + ...

Links

Crossrefs

Cf. A009334, A009353, A009356, A291483, A296435, A296436.

Programs

  • Maple
    a:=series(log(1+arcsinh(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 + ArcSinh[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Log[1 + Log[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 + asinh(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017

Formula

E.g.f.: log(1 + log(x + sqrt(1 + x^2)))*exp(x).
a(n) ~ n! * 2*sqrt(2/Pi) * (Pi*c - 2*s) / (n^(3/2) * (4 + Pi^2)) * (1 + (c*(-192 + 208*Pi - 96*Pi^2 - 8*Pi^3 - 12*Pi^4 + Pi^5) - 2*s*(80 + 48*Pi - 40*Pi^2 + 24*Pi^3 + Pi^4 + 3*Pi^5)) / (4*(4 + Pi^2)^2 * (c*Pi - 2*s)*n)), where s = sin(1 - Pi*n/2) and c = cos(1 - Pi*n/2). - Vaclav Kotesovec, Dec 21 2017

A297213 Expansion of e.g.f. log(1 + arctanh(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 10, -40, 213, -1383, 11002, -100616, 1062625, -12508067, 164543938, -2368224032, 37311284645, -634900302775, 11658800863330, -229004281334768, 4804124787023265, -106986109080667043, 2524701174424967130, -62860054802079553016, 1648303843512405478485
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			log(1 + arctanh(x))*exp(-x) = x/1! - 3*x^2/2! + 10*x^3/3! - 40*x^4/4! + 213*x^5/5! - 1383*x^6/6! + ...

Links

Crossrefs

Cf. A009337, A009356, A009374, A009393, A202139, A296439, A297209, A297210, A297211.

Programs

  • Maple
    S:= series(log(1+arctanh(x))*exp(-x),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Jul 09 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[Log[1 + ArcTanh[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Log[1 + (Log[1 + x] - Log[1 - x])/2] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

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

A297210 Expansion of e.g.f. log(1 + arcsinh(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 7, -16, 48, -213, 1027, -4856, 32512, -309377, 2527963, -16805072, 179877332, -2916171997, 32511289795, -227822369168, 3575741575680, -98643332014049, 1352701143217491, -6534261348983096, 168508582018012980, -9094443640555413357, 143341194607564099595
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			log(1 + arcsinh(x))*exp(-x) = x/1! - 3*x^2/2! + 7*x^3/3! - 16*x^4/4! + 48*x^5/5! - 213*x^6/6! + ...

Crossrefs

Cf. A009337, A009356, A009374, A009393, A296435, A296437, A297209, A297211, A297213.

Programs

  • Maple
    a:=series(log(1+arcsinh(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 + ArcSinh[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + Log[x + Sqrt[1 + x^2]]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

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]!
Showing 1-5 of 5 results.

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