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.

A360601 E.g.f. satisfies A(x) = exp(x*A(x)^2) / (1-x).

Original entry on oeis.org

1, 2, 13, 166, 3265, 87306, 2957509, 121400350, 5857287937, 324884241874, 20370279663901, 1424790170536470, 109990236302275201, 9289460282062082266, 852049115732672006101, 84345608594930495005966, 8962937531710834906989313, 1017655033307013508626619554
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2023

Keywords

Links

Crossrefs

Cf. A352410, A360609.
Cf. A052750, A352448.
Cf. A370875.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(lambertw(-2*x/(1-x)^2)/(-2*x))))

Formula

E.g.f.: sqrt(LambertW( -2*x/(1-x)^2 ) / (-2*x)).
a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 2*exp(-1))) * n^(n-1) / (2 * (sqrt(1 + 2*exp(-1)) - 1)^(3/2) * exp(2*n + 1/2) * (1 + exp(-1) - sqrt(1 + 2*exp(-1)))^n). - Vaclav Kotesovec, Mar 06 2023
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n+k,n-k)/k!. - Seiichi Manyama, Mar 09 2024

A370876 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^3)) ).

Original entry on oeis.org

1, 1, 2, 12, 120, 1320, 17640, 304920, 6249600, 143579520, 3711052800, 107762054400, 3455138332800, 120802387305600, 4583177081683200, 187766031131078400, 8256125218115174400, 387662886088250572800, 19364540503274942976000, 1025507260911983244595200
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2024

Keywords

Links

Crossrefs

Cf. A352410, A370875.
Cf. A360609.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(x+exp(x^3)))/x))
  • PARI
    a(n) = n!*sum(k=0, n\3, (3*k+1)^(k-1)*binomial(n, 3*k)/k!);

Formula

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

A371040 E.g.f. satisfies A(x) = exp(x^3*A(x)^2) / (1-x).

Original entry on oeis.org

1, 1, 2, 12, 96, 840, 9720, 143640, 2399040, 45239040, 976752000, 23537606400, 621444700800, 17936155036800, 562855739846400, 19038932398886400, 690456599575142400, 26748823900403404800, 1102407824344284057600, 48147134965603914240000
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2024

Keywords

Links

Crossrefs

Cf. A360601, A370875.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sqrt(lambertw(-2*x^3/(1-x)^2)/(-2*x^3))))
  • PARI
    a(n) = n!*sum(k=0, n\3, (2*k+1)^(k-1)*binomial(n-k, n-3*k)/k!);

Formula

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