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.

A351274 a(0) = 1; thereafter a(n) = Sum_{k=1..n} (2*k)^k * Stirling1(n,k).

Original entry on oeis.org

1, 2, 14, 172, 2964, 65848, 1789688, 57521280, 2133964352, 89744964288, 4219022123328, 219246630903936, 12479659844383104, 772174659456713472, 51603153976362554112, 3704166182571098222592, 284239227254465994240000, 23218955083323248158556160
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A305819, A351182, A351275, A351276.

Programs

  • Mathematica
    Join[{1},Table[Sum[(2k)^k StirlingS1[n,k],{k,n}],{n,20}]] (* Harvey P. Dale, Dec 31 2023 *)
  • 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) ~ n^n / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n+1/2) * exp(n - exp(-1)/4 + 1/2)). - Vaclav Kotesovec, Feb 06 2022

Extensions

Made a(0) = 1 explicit and changed range of k in definition to start at 1 at the suggestion of Harvey P. Dale. - N. J. A. Sloane, Dec 31 2023

A351277 a(n) = Sum_{k=0..n} (-2*k)^k * Stirling2(n,k).

Original entry on oeis.org

1, -2, 14, -170, 2910, -64202, 1733278, -55338250, 2039421598, -85204516298, 3979272245662, -205432301027978, 11616783053131934, -714082744228546890, 47409028234931260318, -3380871137079666543114, 257736986308003127354014
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A351218, A351275, A351276.

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*(exp(x)-1)))))

Formula

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

A355779 E.g.f. satisfies A(x) = 1/(1 - x)^(2 * A(x)).

Original entry on oeis.org

1, 2, 14, 168, 2912, 66600, 1900056, 65101120, 2606993728, 119561789952, 6181730106240, 355838533286016, 22573258090527360, 1564818434983235328, 117698836976753297664, 9547346757806586746880, 830846347686871026714624, 77215374643802544102187008
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Crossrefs

Cf. A052813, A351275, A355786.

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)*abs(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 18 2022: (Start)
E.g.f.: LambertW(2 * log(1-x)) / (2 * log(1-x)).
a(n) ~ sqrt(2) * n^(n-1) * exp(3/2 - n + n*exp(-1)/2) / (exp(exp(-1)/2) - 1)^(n - 1/2). (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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