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 1-4 of 4 results.

A362571 E.g.f. satisfies A(x) = exp(x * A(x)^(x^2)).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 8401, 82321, 456625, 11496241, 172149121, 1452983401, 40947003241, 823437038425, 9491714865361, 300842942443681, 7568303382376801, 111494036396244961, 3957438528527140225, 119206427681076135481, 2147109997071581380441
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2023

Keywords

Links

Crossrefs

Cf. A000272, A361777.
Cf. A362569, A362573.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-lambertw(-x^3)))))

Formula

E.g.f.: (-LambertW(-x^3) / x^3)^(1/x^2) = exp(-LambertW(-x^3) / x^2) = exp(x * exp(-LambertW(-x^3))).
a(n) = n! * Sum_{k=0..floor(n/3)} (n-2*k)^k * binomial(n-2*k-1,k)/(n-2*k)!.
E.g.f.: Sum_{k>=0} (k*x^2 + 1)^(k-1) * x^k / k!.

A362799 E.g.f. satisfies A(x) = exp( (exp(x) - 1) * A(x)^x ).

Original entry on oeis.org

1, 1, 2, 11, 63, 542, 5183, 62211, 830252, 12900381, 220566835, 4223662522, 88001471869, 2007052809465, 49309469989666, 1306455781607975, 36973887007453315, 1116728635342926570, 35775769695237122035, 1213704083311914974899
Offset: 0

Views

Author

Seiichi Manyama, May 04 2023

Keywords

Links

Crossrefs

Cf. A052880, A362800.
Cf. A362794, A362796.
Cf. A000110, A361777.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(exp(x)-1))/x)))

Formula

E.g.f.: exp( -LambertW(-x * (exp(x) - 1)) / x ).
E.g.f.: Sum_{k>=0} (k*x + 1)^(k-1) * (exp(x) - 1)^k / k!.

A362568 E.g.f. satisfies A(x) = exp(x/A(x)^x).

Original entry on oeis.org

1, 1, 1, -5, -23, 121, 1321, -7349, -148175, 853777, 27840241, -163354949, -7934320679, 46820981065, 3203091569497, -18833438286389, -1742847946697759, 10137524365568161, 1230956201929018465, -7042544858204663813, -1095864481054115534519
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2023

Keywords

Links

Crossrefs

Cf. A177885, A362569.
Cf. A361777.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-lambertw(x^2)))))

Formula

E.g.f.: (x^2 / LambertW(x^2))^(1/x) = exp(LambertW(x^2) / x) = exp(x * exp(-LambertW(x^2))).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * (n-k)^k * binomial(n-k-1,k)/(n-k)!.
E.g.f.: Sum_{k>=0} (-k*x + 1)^(k-1) * x^k / k!.

A362572 E.g.f. satisfies A(x) = exp(x * A(x)^(x/2)).

Original entry on oeis.org

1, 1, 1, 4, 13, 76, 421, 3361, 26209, 267688, 2689201, 33579811, 412800961, 6103089994, 88754687113, 1517513934301, 25487131948321, 495009722435176, 9430633148123809, 205154208873930763, 4371962638221712801, 105330237499426955926
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2023

Keywords

Links

Crossrefs

Cf. A000272, A362573.
Cf. A361777.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-lambertw(-x^2/2)))))

Formula

E.g.f.: (-2 * LambertW(-x^2/2) / x^2)^(2/x) = exp(-2 * LambertW(-x^2/2) / x) = exp(x * exp(-LambertW(-x^2/2))).
a(n) = n! * Sum_{k=0..floor(n/2)} ((n-k)/2)^k * binomial(n-k-1,k)/(n-k)!.
E.g.f.: Sum_{k>=0} (k*x/2 + 1)^(k-1) * x^k / k!.
a(n) ~ (exp(sqrt(2)*exp(1/2)) - (-1)^n*exp(-sqrt(2)*exp(1/2))) * n^(n-1) / (2^(n/2 - 1) * exp((n-1)/2)). - Vaclav Kotesovec, Aug 05 2025
Showing 1-4 of 4 results.

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