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.

A352139 Expansion of e.g.f. 1/(exp(x) - log(1 - x)).

Original entry on oeis.org

1, -2, 6, -27, 161, -1205, 10799, -113043, 1351461, -18183781, 271784079, -4469044657, 80160267791, -1557710354083, 32597642189657, -730897865864471, 17480390183397209, -444198879957594857, 11951585821669838395, -339434402344422296117
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 19; 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, ((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).

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).

A352295 Expansion of e.g.f. 1/(exp(x) - x/(1 + x)).

Original entry on oeis.org

1, 0, -3, 5, 29, -181, -401, 9645, -14183, -689257, 4826171, 55700633, -1024570955, -2770525005, 221566919911, -1028838834811, -49439771820367, 723165789334703, 9903852025111027, -362150510124039471, -463774017017434739, 169793689786411161995
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x/(1 + x)), {x, 0, m}], x] (* Amiram Eldar, Mar 11 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x/(1+x))))
    
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, ((-1)^(k-1)*k!-1)*binomial(n, k)*a(n-k)));

Formula

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