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

A355780 E.g.f. satisfies A(x) = (1 + x)^(2 * A(x)).

Original entry on oeis.org

1, 2, 10, 96, 1352, 25400, 597816, 16941568, 561993344, 21372060672, 916910785920, 43817650647936, 2308500130055808, 132941831957885184, 8308594453077321984, 560108109905112238080, 40514005700203717945344, 3129925644058623770173440
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Crossrefs

Cf. A033917, A351274, A355787.

Programs

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

Formula

E.g.f.: exp( -LambertW(-2 * log(1+x)) ).
a(n) = Sum_{k=0..n} 2^k * (k+1)^(k-1) * Stirling1(n,k).
From Vaclav Kotesovec, Jul 17 2022: (Start)
E.g.f.: -LambertW(-2*log(1+x)) / (2*log(1+x)).
a(n) ~ sqrt(2) * n^(n-1) / ((exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n - 3/2 + exp(-1)/4)). (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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