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.

A351931 Expansion of e.g.f. exp(x - x^5/120).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, -5, -20, -55, -125, -125, 925, 7525, 34750, 124125, 249250, -1013375, -14708875, -97413875, -477236375, -1443329375, 3466472500, 91499089375, 804081585000, 5030009685625, 20366827624375, -23484049500625, -1391395435656875, -15503027252406875
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A351929, A351930.
Cf. A275423, A351906.

Programs

  • Mathematica
    m = 28; Range[0, m]! * CoefficientList[Series[Exp[x - x^5/5!], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x-x^5/5!)))
  • PARI
    a(n) = n!*sum(k=0, n\5, (-1/5!)^k*binomial(n-4*k, k)/(n-4*k)!);
  • PARI
    a(n) = if(n<5, 1, a(n-1)-binomial(n-1, 4)*a(n-5));

Formula

a(n) = n! * Sum_{k=0..floor(n/5)} (-1/5!)^k * binomial(n-4*k,k)/(n-4*k)!.
a(n) = a(n-1) - binomial(n-1,4) * a(n-5) for n > 4.

A351905 Expansion of e.g.f. exp(x * (1 - x^3)).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, -839, 18481, 178417, 902161, 3318481, -69866279, -1011908039, -7204341143, -36194591159, 726745175521, 14326789219681, 131901636673441, 840736509931297, -16060449291985079, -408041402342457239, -4618341644958693959, -35691963052019431079
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2022

Keywords

Crossrefs

Cf. A293604, A246607, A351906.
Cf. A190875, A293493.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x^3))))
  • PARI
    a(n) = n!*sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)/(n-3*k)!);
  • PARI
    a(n) = if(n<4, 1, a(n-1)-4!*binomial(n-1, 3)*a(n-4));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k)/(n-3*k)!.
a(n) = a(n-1) - 4! * binomial(n-1,3) * a(n-4) for n > 3.

A362324 a(n) = n! * Sum_{k=0..floor(n/5)} (-n)^k / (k! * (n-5*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, -599, -4319, -17639, -53759, -136079, 181137601, 2414356561, 17242917121, 87695201881, 355974659041, -734340892685399, -14279571631503359, -145614163414530719, -1037158816523518079, -5794132157196668639, 16192314610730781350401
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2023

Keywords

Links

Crossrefs

Cf. A362282, A362305, A362322.
Cf. A351906, A362320.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((lambertw(5*x^5)/5)^(1/5))/(1+lambertw(5*x^5))))

Formula

a(n) = n! * [x^n] exp(x - n*x^5).
E.g.f.: exp( ( LambertW(5*x^5)/5 )^(1/5) ) / (1 + LambertW(5*x^5)).
Showing 1-3 of 3 results.

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

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