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.

A361194 E.g.f. satisfies A(x) = exp( -3*x*A(x) ) / (1-x).

Original entry on oeis.org

1, -2, 17, -237, 4893, -133683, 4567905, -187666587, 9017657433, -496470972951, 30824023641669, -2131090659947439, 162397790115179733, -13525005928296072915, 1222285110682680848169, -119135392516302191619507, 12458374493322416970025521
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2023

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-3)^k * (k+1)^(k-1) * binomial(n,k)/k!.
E.g.f.: LambertW( 3*x/(1-x) ) / (3*x).

A361213 E.g.f. satisfies A(x) = exp( 2*x*A(x) / (1+x) ).

Original entry on oeis.org

1, 2, 8, 68, 848, 14192, 298048, 7546016, 223792640, 7612381952, 292216807424, 12497875215872, 589392367925248, 30386736933804032, 1700376343771136000, 102641314849948602368, 6648428846464054919168, 459977466799800897437696
Offset: 0

Views

Author

Seiichi Manyama, Mar 04 2023

Keywords

Crossrefs

Programs

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

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} (-2)^k * (k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp ( -LambertW(-2*x/(1+x)) ).
E.g.f.: -(1+x)/(2*x) * LambertW(-2*x/(1+x)).
a(n) ~ (2*exp(1) - 1)^(n + 1/2) * n^(n-1) / (sqrt(2) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 10 2023
Showing 1-2 of 2 results.