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.

A385424 Expansion of e.g.f. exp( -LambertW(-arcsin(x)) ).

Original entry on oeis.org

1, 1, 3, 17, 137, 1465, 19499, 311873, 5829073, 124796081, 3012319315, 80960234577, 2398138520409, 77630951407529, 2726829925494011, 103300796618253825, 4198494172961579169, 182239547736082960737, 8414068749731088539299, 411754575622058760824593
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Links

Crossrefs

Cf. A385426, A385427.
Cf. A277502, A381142, A385343, A385425.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-asin(x)))))

Formula

E.g.f. A(x) satisfies A(x) = exp( arcsin(x) * A(x) ).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A385343(n,k).
a(n) ~ n^(n-1) / (sqrt(cos(exp(-1))) * sin(exp(-1))^(n - 1/2) * exp(n - 3/2)). - Vaclav Kotesovec, Jun 28 2025

A385428 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 0, -11, -80, -219, 3416, 68265, 550656, -3285975, -194101248, -3177823395, -5431320960, 1202586098637, 35658624599040, 359507959906641, -12186663090266112, -677861502762897711, -13768767870225444864, 126162451289700276165, 19553934035547470168064
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Cf. A001147, A385425.
Cf. A381147, A385343, A385369.

Programs

  • PARI
    a385343(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);
a(n) = sum(k=0, n, (n-k+1)^(k-1)*I^(n-k)*a385343(n, k));

Formula

E.g.f. A(x) satisfies A(x) = ( x*A(x) + sqrt((x*A(x))^2 + 1) )^(1/A(x)).
a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.

A385500 Expansion of e.g.f. exp( -LambertW(-arctanh(x)) ).

Original entry on oeis.org

1, 1, 3, 18, 149, 1640, 22359, 366128, 6998697, 153191808, 3779353515, 103800229632, 3141633970749, 103904351855616, 3728602377979647, 144297781732300800, 5991021498320041809, 265636734347975688192, 12527923794824003280723, 626224876080360687599616
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2025

Keywords

Links

Crossrefs

Cf. A385501, A385502.
Cf. A111594, A385425.

Programs

  • Mathematica
    nmax=19; CoefficientList[Series[Exp[ -LambertW[-ArcTanh[x]]],{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Jul 01 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-atanh(x)))))

Formula

E.g.f. A(x) satisfies A(x) = exp( arctanh(x) * A(x) ).
E.g.f. A(x) satisfies A(x) = ( (1+x)/(1-x) )^(A(x)/2).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A111594(n,k).
a(n) ~ sqrt(exp(4*exp(-1)) - 1) * n^(n-1) / (2*exp(n - 3/2 + exp(-1)) * tanh(exp(-1))^n). - Vaclav Kotesovec, Jul 01 2025
Showing 1-3 of 3 results.

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

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