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.

A372315 Expansion of e.g.f. exp( x - LambertW(-2*x)/2 ).

Original entry on oeis.org

1, 2, 8, 68, 960, 18832, 471136, 14324480, 512733696, 21119803136, 984029612544, 51169331031040, 2937675286583296, 184560174104465408, 12594824112085327872, 927757127285523243008, 73369903633161123397632, 6200198958236463387836416
Offset: 0

Views

Author

Seiichi Manyama, Apr 27 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-2*x)/2)))
    
  • PARI
    a(n) = sum(k=0, n, (2*k+1)^(k-1)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ 2^(n-1) * n^(n-1) * exp((exp(-1) + 1)/2). - Vaclav Kotesovec, May 04 2024

A372320 Expansion of e.g.f. -exp( x + LambertW(-2*x)/2 ).

Original entry on oeis.org

-1, 0, 4, 36, 464, 8560, 206112, 6104896, 214376192, 8701657344, 400748710400, 20642974511104, 1175888936749056, 73389707156586496, 4980134850525986816, 365062349226075463680, 28747688571714736160768, 2420266280392895064506368
Offset: 0

Views

Author

Seiichi Manyama, Apr 27 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x+lambertw(-2*x)/2)))
    
  • PARI
    a(n) = sum(k=0, n, (2*k-1)^(k-1)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} (2*k-1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (2*k-1)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ 2^(n-1) * n^(n-1) * exp((exp(-1) - 1)/2). - Vaclav Kotesovec, May 06 2024

A372321 Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).

Original entry on oeis.org

-1, 0, 6, 81, 1620, 45765, 1671678, 74794671, 3958829640, 241898775273, 16756621904970, 1297547591499819, 111065107263415308, 10412999996499836541, 1061234184094567585326, 116812280111404106348415, 13810631408232372091755792, 1745470697932523785587735249
Offset: 0

Views

Author

Seiichi Manyama, Apr 27 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x+lambertw(-3*x)/3)))
    
  • PARI
    a(n) = sum(k=0, n, (3*k-1)^(k-1)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} (3*k-1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (3*k-1)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ 3^(n-1) * n^(n-1) * exp((exp(-1) - 1)/3). - Vaclav Kotesovec, May 06 2024

A372334 Expansion of e.g.f. -exp(x) * LambertW(-3*x)/3.

Original entry on oeis.org

0, 1, 8, 102, 2092, 60140, 2220954, 100119670, 5328468968, 326960686872, 22724388453070, 1764411577328906, 151364204180518476, 14217940294767407380, 1451334877597451677250, 159972528561402504191190, 18936257811933773637390544, 2395818853376147403857700656
Offset: 0

Views

Author

Seiichi Manyama, Apr 28 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-exp(x)*lambertw(-3*x)/3)))
    
  • PARI
    a(n) = sum(k=1, n, (3*k)^(k-1)*binomial(n, k));

Formula

a(n) = Sum_{k=1..n} (3*k)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=1} (3*k)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ exp(exp(-1)/3) * 3^(n-1) * n^(n-1). - Vaclav Kotesovec, Apr 30 2024
Showing 1-4 of 4 results.