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 1-4 of 4 results.

A368267 Expansion of e.g.f. exp(2*x) / (1 - 2*x*exp(x)).

Original entry on oeis.org

1, 4, 24, 206, 2344, 33322, 568420, 11312366, 257293872, 6583516946, 187173328444, 5853594770806, 199705444781512, 7381068971010074, 293787494031046740, 12528831526596461438, 569923490454177217120, 27545552296682691393058
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, 2^(n-k)*(n-k+2)^k/k!);

Formula

a(n) = n! * Sum_{k=0..n} 2^(n-k) * (n-k+2)^k / k!.
a(n) ~ n! / (4 * LambertW(1/2)^(n+2) * (LambertW(1/2) + 1)). - Vaclav Kotesovec, Dec 29 2023

A368268 Expansion of e.g.f. exp(-x) / (1 - 2*x*exp(x)).

Original entry on oeis.org

1, 1, 9, 71, 817, 11599, 197881, 3938087, 89569761, 2291869727, 65159228521, 2037767466679, 69521938950289, 2569515452879855, 102274007835523161, 4361566914028222919, 198403133940750790081, 9589223805173365594687, 490729273233730201604809
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, 2^(n-k)*(n-k-1)^k/k!);

Formula

a(n) = n! * Sum_{k=0..n} 2^(n-k) * (n-k-1)^k / k!.

A368269 Expansion of e.g.f. exp(-2*x) / (1 - 2*x*exp(x)).

Original entry on oeis.org

1, 0, 8, 46, 584, 8138, 139252, 2770206, 63009648, 1612255186, 45837395564, 1433503025414, 48906419204392, 1807570412699322, 71946432680652324, 3068220235065662062, 139570141248903198944, 6745706553985526731682, 345212056986241161670876
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, 2^(n-k)*(n-k-2)^k/k!);

Formula

a(n) = n! * Sum_{k=0..n} 2^(n-k) * (n-k-2)^k / k!.

A368237 Expansion of e.g.f. 1/(exp(-x) - 3*x).

Original entry on oeis.org

1, 4, 31, 361, 5605, 108781, 2533447, 68836279, 2137543177, 74673228457, 2898494302651, 123757822391083, 5764497138070381, 290878956151681405, 15806942065094830735, 920336494043393536591, 57157621592164505969425, 3771643127452655490322513
Offset: 0

Views

Author

Seiichi Manyama, Dec 18 2023

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := n! Sum[3^(n - k) (n - k + 1)^k / k!, {k, 0, n}];Table[a[n],{n,0,17}] (* or *) a[0] = 1;a[n_] := 3n a[n - 1] + Sum[(-1)^(k - 1) Binomial[n, k] a[n - k], {k, 1, n}];Table[a[n],{n,0,17}] (* James C. McMahon, Dec 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, 3^(n-k)*(n-k+1)^k/k!);
    
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(exp(-x) - 3*x))) \\ Michel Marcus, Dec 18 2023

Formula

a(0) = 1; a(n) = 3*n*a(n-1) + Sum_{k=1..n} (-1)^(k-1) * binomial(n,k) * a(n-k).
a(n) = n! * Sum_{k=0..n} 3^(n-k) * (n-k+1)^k / k!.
Showing 1-4 of 4 results.