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.

A372316 Expansion of e.g.f. exp( x - LambertW(-3*x)/3 ).

Original entry on oeis.org

1, 2, 10, 125, 2644, 77597, 2904382, 132169403, 7083715240, 437031850841, 30506442905194, 2377038378159359, 204521399708464252, 19259006462435865413, 1970114326513629358654, 217556451608123850352523, 25794252755430105917806288, 3268152272130255473300883377
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 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

A372333 Expansion of e.g.f. -exp(x) * LambertW(-2*x)/2.

Original entry on oeis.org

0, 1, 6, 51, 684, 12965, 317298, 9500631, 336237016, 13729172553, 635237632350, 32844916975739, 1876755685038468, 117437155609780461, 7986793018367861194, 586578825469711599135, 46268265552518066488752, 3901008402618593931019409
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(-2*x)/2)))
    
  • PARI
    a(n) = sum(k=1, n, (2*k)^(k-1)*binomial(n, k));

Formula

a(n) = Sum_{k=1..n} (2*k)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=1} (2*k)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ exp(exp(-1)/2) * 2^(n-1) * n^(n-1). - Vaclav Kotesovec, Apr 30 2024
Showing 1-4 of 4 results.