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 1-2 of 2 results.

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)

A356909 E.g.f. satisfies A(x)^A(x) = 1/(1 - 2*x)^(1/2).

Original entry on oeis.org

1, 1, 1, 6, 25, 260, 2031, 29414, 343729, 6331464, 99347775, 2213854962, 43775364777, 1143045179484, 27318538754703, 817488701386590, 22930078960458081, 773538392955105552, 24914603847636688383, 935548306987015373418, 34024565857537814925465
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A355786, A356908.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = (1 - 2*x)^(-(1/2)/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*(-k+1)^(k-1)*abs(stirling(n, k, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-log(1-2*x)/2)^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-log(1-2*x)/2))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-log(1-2*x)/(2*lambertw(-log(1-2*x)/2))))

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * (-k+1)^(k-1) * |Stirling1(n,k)|.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-log(1-2*x)/2)^k / k!.
E.g.f.: A(x) = exp( LambertW(-log(1-2*x)/2) ).
E.g.f.: A(x) = -log(1-2*x)/(2 * LambertW(-log(1-2*x)/2)).
Showing 1-2 of 2 results.

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

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