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.

A356911 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, -12096, -120960, -1144800, -11642400, 190270080, 4670265600, 81378198720, 1348668921600, -880532674560, -406217626214400, -13255586359142400, -343166884178227200, -3137937973466572800, 72862796986940620800
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A356905, A356910.
Cf. A353229, A356099.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^3))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (-k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^3*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3*log(1-x)/lambertw(-x^3*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-k+1)^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^3 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^3 * log(1-x)) ).
E.g.f.: A(x) = -x^3 * log(1-x)/LambertW(-x^3 * log(1-x)).

A356950 E.g.f. satisfies log(A(x)) = x^3 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 60816, 544824, 3175920, 14969790, 1339209960, 25141598196, 291418089144, 2618105492730, 128974591028640, 3841451570440560, 73103023032142176, 1060951475511351414, 39132892925113341240, 1516348247446904304300
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356949.
Cf. A000272, A356099, A356952.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])*A[x]*x^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*stirling(n-3*k, k, 2)/(n-3*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^3*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3*(1-exp(x))))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3*(1-exp(x)))/(x^3*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * Stirling2(n-3*k,k)/(n-3*k)!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^3 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^3 * (1 - exp(x)))/(x^3 * (1 - exp(x))).
Showing 1-2 of 2 results.

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