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.

A371371 E.g.f. satisfies A(x) = exp(x/(1 - A(x))^2) - 1.

Original entry on oeis.org

0, 1, 5, 61, 1209, 33261, 1171933, 50363293, 2554659761, 149399423101, 9896519640981, 732401926901613, 59890184672573929, 5362586032967290765, 521831581416561627149, 54834132144912233219581, 6188110724712474697469025, 746431260858514472012500701
Offset: 0

Views

Author

Seiichi Manyama, Mar 20 2024

Keywords

Crossrefs

Programs

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

Formula

E.g.f.: Series_Reversion( (1 - x)^2 * log(1+x) ).
a(n) = Sum_{k=1..n} (2*n+k-2)!/(2*n-1)! * Stirling2(n,k).
a(n) ~ 2^(n-1) * LambertW(exp(1/2))^(2*n-1) * n^(n-1) / (sqrt(LambertW(exp(1/2)) + 1) * exp(n) * (2*LambertW(exp(1/2))-1)^(3*n-1)). - Vaclav Kotesovec, Mar 29 2024

A376040 E.g.f. satisfies A(x) = -log(1 - x / (1 - A(x))^3).

Original entry on oeis.org

0, 1, 7, 119, 3258, 123414, 5978082, 352880562, 24573720672, 1972239280488, 179250831525720, 18197871488362752, 2041093578923498448, 250654006995798120480, 33449544716000374458000, 4819960747934844400104480, 745867334512204468875843840
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(serreverse((1-x)^3*(1-exp(-x))))))
    
  • PARI
    a(n) = sum(k=1, n, (3*n+k-2)!/(3*n-1)!*abs(stirling(n, k, 1)));

Formula

E.g.f.: Series_Reversion( (1 - x)^3 * (1 - exp(-x)) ).
a(n) = Sum_{k=1..n} (3*n+k-2)!/(3*n-1)! * |Stirling1(n,k)|.

A368033 E.g.f. satisfies A(x) = log(1 + x/(1 - A(x))^2).

Original entry on oeis.org

0, 1, 3, 26, 370, 7334, 186468, 5787144, 212100208, 8964974016, 429304991880, 22971063265776, 1358260804832160, 87949592273821680, 6189420503357272608, 470384337802047909120, 38393707193347187344896, 3349704214386311986028160
Offset: 0

Views

Author

Seiichi Manyama, Mar 20 2024

Keywords

Crossrefs

Programs

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

Formula

E.g.f.: Series_Reversion( (1 - x)^2 * (exp(x) - 1) ).
a(n) = Sum_{k=1..n} (2*n+k-2)!/(2*n-1)! * Stirling1(n,k).
a(n) ~ LambertW(2*exp(1))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(1)))) * exp(n) * (2 - LambertW(2*exp(1)))^(3*n - 1)). - Vaclav Kotesovec, Mar 29 2024

A376067 E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^2)) * (1 - A(x)).

Original entry on oeis.org

0, 1, 3, 26, 372, 7424, 190150, 5946576, 219643592, 9357076704, 451643892408, 24359462797680, 1451906224395792, 94769186402062080, 6723078079388867040, 515064037555614081024, 42380187502270667120640, 3727409807764337879016960
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=1, n, (2*n-2)!/(2*n-k-1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} (2*n-2)!/(2*n-k-1)! * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( (1 - x)^2 * (1 - exp(-x / (1 - x))) ).
Showing 1-4 of 4 results.