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 1-3 of 3 results.

A295267 Expansion of e.g.f. 2/(1 + sqrt(1 + 4*LambertW(-x))).

Original entry on oeis.org

1, 1, 6, 63, 952, 18885, 465696, 13764667, 475039104, 18767660553, 835805555200, 41442148754391, 2264776308946944, 135268340058044557, 8767315076546568192, 612911076907734961875, 45973645939542007054336, 3683096368557198711874833, 313878687736263437290438656
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 19 2017

Keywords

Links

Crossrefs

Cf. A000108, A180680, A277184, A295268.

Programs

  • Maple
    a:=series(2/(1+sqrt(1+4*LambertW(-x))),x=0,19): seq(n!*coeff(a,x,n),n=0..18); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 18; CoefficientList[Series[2/(1 + Sqrt[1 + 4 LambertW[-x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[LambertW[-x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    x ='x+O('x^30); Vec(serlaplace(2/(1 +sqrt(1 +4*lambertw(-x))))) \\ G. C. Greubel, Jul 07 2018

Formula

E.g.f.: 1/(1 + LambertW(-x)/(1 + LambertW(-x)/(1 + LambertW(-x)/(1 + LambertW(-x)/(1 + ...))))), a continued fraction.
a(n) ~ 2^(2*n + 3/2) * n^(n-1) / (sqrt(3) * exp(3*n/4)). - Vaclav Kotesovec, Nov 19 2017

A295268 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*LambertW(x))).

Original entry on oeis.org

1, 1, 2, 15, 104, 1445, 18144, 364651, 6761600, 176898249, 4376614400, 140703601511, 4370369292288, 166520945009965, 6235421191430144, 274675339364551875, 12046634866183798784, 602474837696641959569, 30289753591657339944960, 1696072847731424941183039
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 19 2017

Keywords

Links

Crossrefs

Cf. A000108, A180680, A277184, A295267.

Programs

  • Maple
    S:= series(2/(1 + sqrt(1 - 4*LambertW(x))), x, 31):
seq(coeff(S,x,n)*n!,n=0..30); # Robert Israel, Nov 20 2017
  • Mathematica
    nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 - 4 LambertW[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-LambertW[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    x = 'x + O('x^30); Vec(serlaplace(2/(1 + sqrt(1 - 4*lambertw(x))))) \\ Michel Marcus, Nov 20 2017

Formula

E.g.f.: 1/(1 - LambertW(x)/(1 - LambertW(x)/(1 - LambertW(x)/(1 - LambertW(x)/(1 - ...))))), a continued fraction.
a(n) ~ 2^(2*n + 3/2) * n^(n-1) / (sqrt(5) * exp(5*n/4)). - Vaclav Kotesovec, Nov 19 2017

A380673 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x * (1 - x)) ).

Original entry on oeis.org

1, 2, 11, 106, 1501, 28416, 677839, 19566128, 663801849, 25897000960, 1142424023731, 56232973813248, 3055417111781269, 181644488496644096, 11728204122824976375, 817281148114199197696, 61136484485752079320561, 4886365932210442324672512, 415573028022035962921316059
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Links

Crossrefs

Cf. A377831, A380663, A380664.
Cf. A277184.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*n-2*k, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x) * (1 - x*A(x)))/(1 - x*A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n-2*k,n-k)/k!.
a(n) = A277184(n+1)/(n+1).
Showing 1-3 of 3 results.

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