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.

A356908 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^2.

Original entry on oeis.org

1, 2, -2, 24, -272, 4840, -107496, 2934400, -94501760, 3511914624, -147882696960, 6959217277056, -361941871760256, 20616017227643136, -1276341094954066176, 85337532623368181760, -6128269375791673718784, 470426144496265208979456
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A349561, A355779, A356909.

Programs

  • Maple
    S:= series(exp( LambertW(-2 * log(1-x))),x,51):
seq(n!*coeff(S,x,n),n=0..50); # Robert Israel, Sep 18 2022
  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1-x)^2)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    a(n) = sum(k=0, n, 2^k*(-k+1)^(k-1)*abs(stirling(n, k, 1)));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-2*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-2*log(1-x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-2*log(1-x)/lambertw(-2*log(1-x))))

Formula

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

A356912 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2/2).

Original entry on oeis.org

1, 0, 0, 3, 6, 20, 0, -126, -1260, 18360, 335160, 4546080, 26302320, -59501520, -5703994296, -58549768200, 371346066000, 34962417322560, 746101280831040, 8059680118183680, -93772611412099200, -5613314502242643840, -110940169654432087200
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A351492, A356752.
Cf. A356909, A356910.

Programs

  • Mathematica
    nmax = 22; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^2/2))^(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\3, (-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^2/2*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^2/2*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^2/2*log(1-x)/lambertw(-x^2/2*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^2/2 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^2/2 * log(1-x)) ).
E.g.f.: A(x) = -x^2/2 * log(1-x)/LambertW(-x^2/2 * log(1-x)).
Showing 1-2 of 2 results.

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