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.

A362569 E.g.f. satisfies A(x) = exp(x/A(x)^(x^2)).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, 6721, 78961, 450577, -7867439, -160506719, -1421049959, 23995634521, 745945175977, 9197488067041, -152057966904479, -6667968305775839, -107047941299543519, 1740437689443523777, 102311231044267813321, 2043217889363061489961
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2023

Keywords

Links

Crossrefs

Cf. A177885, A362568.
Cf. A362571.

Programs

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

Formula

E.g.f.: (x^3 / LambertW(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)} (-1)^k * (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!.

A362573 E.g.f. satisfies A(x) = exp(x * A(x)^(x^2/6)).

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 351, 2521, 13105, 96041, 933021, 7098301, 65348141, 787190405, 7896243811, 88712631281, 1269172794401, 15784837036561, 210688183375705, 3486485630182581, 51674172769168741, 801474314335394701, 15059801657898920231, 258815184609843935305
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2023

Keywords

Links

Crossrefs

Cf. A000272, A362572.
Cf. A362571.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-lambertw(-x^3/6)))))

Formula

E.g.f.: (-6 * LambertW(-x^3/6) / x^3)^(6/x^2) = exp(-6 * LambertW(-x^3/6) / x^2) = exp(x * exp(-LambertW(-x^3/6))).
a(n) = n! * Sum_{k=0..floor(n/3)} ((n-2*k)/6)^k * binomial(n-2*k-1,k)/(n-2*k)!.
E.g.f.: Sum_{k>=0} (k*x^2/6 + 1)^(k-1) * x^k / k!.

A362800 E.g.f. satisfies A(x) = exp( (exp(x) - 1) * A(x)^(x^2) ).

Original entry on oeis.org

1, 1, 2, 5, 39, 292, 2063, 21877, 271372, 3298155, 47855035, 805112970, 13843621861, 261388560253, 5529798475178, 122059754102345, 2863956966387107, 73150334575839340, 1961833778207602123, 55184622355007805281, 1656027290812446938492
Offset: 0

Views

Author

Seiichi Manyama, May 04 2023

Keywords

Links

Crossrefs

Cf. A052880, A362799.
Cf. A362795, A362798.
Cf. A000110, A362571.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x^2*(exp(x)-1))/x^2)))

Formula

E.g.f.: exp( -LambertW(-x^2 * (exp(x) - 1)) / x^2 ).
E.g.f.: Sum_{k>=0} (k*x^2 + 1)^(k-1) * (exp(x) - 1)^k / k!.
Showing 1-3 of 3 results.

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

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