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.

A351275 a(n) = Sum_{k=0..n} (-2*k)^k * Stirling1(n,k).

Original entry on oeis.org

1, -2, 18, -268, 5580, -149368, 4887368, -189010176, 8434813760, -426626153664, 24118046539968, -1507010218083456, 103135804627122816, -7672260068001952512, 616407170000568900864, -53192668792451354284032, 4906864974307552234844160
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A351274, A351277.

Programs

  • PARI
    a(n) = sum(k=0, n, (-2*k)^k*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(2*log(1+x)))))

Formula

E.g.f.: 1/(1 + LambertW( 2 * log(1+x) )), where LambertW() is the Lambert W-function.
a(n) ~ (-1)^n * exp(-1/2 - n + n*exp(-1)/2) * n^n / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n+1/2)). - Vaclav Kotesovec, Feb 06 2022

A351276 a(n) = Sum_{k=0..n} (2*k)^k * Stirling2(n,k).

Original entry on oeis.org

1, 2, 18, 266, 5506, 146602, 4772162, 183618794, 8152995138, 410307648938, 23079780216386, 1434953808618090, 97716253164212034, 7233006174407149866, 578233606405444793410, 49651123488091636885994, 4557474786380802233761090, 445324385454834015896585386
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A282190, A308490, A351274, A351277.

Programs

  • PARI
    a(n) = sum(k=0, n, (2*k)^k*stirling(n, k, 2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(2*(1-exp(x))))))

Formula

E.g.f.: 1/(1 + LambertW( 2 * (1 - exp(x)) )), where LambertW() is the Lambert W-function.
a(n) ~ n^n / (sqrt(1 + 2*exp(1)) * (log(exp(1) + 1/2) - 1)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Feb 06 2022

A355782 E.g.f. satisfies log(A(x)) = 2 * (1 - exp(-x)) * A(x).

Original entry on oeis.org

1, 2, 10, 94, 1314, 24494, 572418, 16109678, 530772610, 20049256686, 854425665410, 40560727143534, 2122785621956226, 121440903560075246, 7539867236251002242, 504946360197545803630, 36284349255747713008770, 2784785703026225861819118
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Crossrefs

Cf. A058864, A351277, A355789.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(exp(-x)-1)))))
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*2^k*(k+1)^(k-1)*stirling(n, k, 2));

Formula

E.g.f.: exp( -LambertW(2 * (exp(-x) - 1)) ).
a(n) = Sum_{k=0..n} (-1)^(n-k) * 2^k * (k+1)^(k-1) * Stirling2(n,k).
From Vaclav Kotesovec, Jul 18 2022: (Start)
E.g.f.: LambertW(2 * (exp(-x) - 1)) / (2 * (exp(-x) - 1)).
a(n) ~ sqrt(2*exp(1) - 1) * sqrt(log(2/(2 - exp(-1)))) * n^(n-1) / (exp(n-1) * (log(2/(2*exp(1)-1)) + 1)^n). (End)
Showing 1-3 of 3 results.

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

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