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.

A352146 Expansion of e.g.f. 1/(exp(x) + log(1 - x)).

Original entry on oeis.org

1, 0, 0, 1, 5, 23, 139, 1069, 9365, 90971, 981647, 11697167, 152304591, 2149063421, 32668289913, 532328418153, 9256383832665, 171066343532055, 3348245897484091, 69189708307509195, 1505284330388457451, 34391324279752372105, 823258887611521993045
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 22; Range[0, m]! * CoefficientList[Series[1/(Exp[x] + Log[1 - x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)+log(1-x))))
    
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, ((k-1)!-1)*binomial(n, k)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} ((k-1)! - 1) * binomial(n,k) * a(n-k).
a(n) ~ n! * (1-r) / ((1 - (1-r)*exp(r)) * r^(n+1)), where r = 0.9183335761894542037857295468680123485973875022318007816308... is the root of the equation exp(r) = -log(1-r). - Vaclav Kotesovec, Mar 06 2022

A352147 Expansion of e.g.f. 1/(exp(x) + log(1 + x)).

Original entry on oeis.org

1, -2, 8, -51, 437, -4685, 60299, -905583, 15543989, -300163717, 6440430159, -152007707357, 3913861488767, -109171084473763, 3279401359094041, -105546729767585411, 3623462164916028569, -132169615185372857001, 5104616345453966073403
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 18; Range[0, m]! * CoefficientList[Series[1/(Exp[x] + Log[1 + x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)+log(1+x))))
    
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, ((-1)^k*(k-1)!-1)*binomial(n, k)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} ((-1)^k * (k-1)! - 1) * binomial(n,k) * a(n-k).

A352138 Expansion of e.g.f. 1/(exp(x) - log(1 + x)).

Original entry on oeis.org

1, 0, -2, 1, 17, -17, -401, 817, 16197, -49861, -1123633, 5354787, 105696447, -682603651, -14697824519, 131535803133, 2457119246745, -28321054685609, -572811846560453, 8626026427105983, 146289547341006011, -2784279036040263575, -51756654994427512331
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 22; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - Log[1 + x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-log(1+x))))
    
  • PARI
    a(n) = if(n==0, 1, -sum(k=1, n, ((-1)^k*(k-1)!+1)*binomial(n, k)*a(n-k)));

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} ((-1)^k * (k-1)! + 1) * binomial(n,k) * a(n-k).

A352270 Expansion of e.g.f. 1/(2 - exp(x) + log(1 - x)).

Original entry on oeis.org

1, 2, 10, 75, 751, 9405, 141361, 2478959, 49683047, 1120216645, 28064294201, 773391141325, 23250533411821, 757231705088131, 26558855360366239, 998051946325525971, 40006049065833007891, 1703833370634756077097, 76833773059665726636621
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2022

Keywords

Crossrefs

Programs

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

Formula

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