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 1-3 of 3 results.

A370922 E.g.f. satisfies A(x) = log(1 + x/(1 - A(x)))/(1 - A(x)).

Original entry on oeis.org

0, 1, 3, 29, 444, 9454, 257822, 8576504, 336770592, 15246592440, 781883091672, 44797478362680, 2836034500712256, 196601715537070752, 14811696896760459264, 1205008924460733794688, 105284627507520312994560, 9832559605580777568425856
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=1..n} (n+2*k-2)!/(n+k-1)! * Stirling1(n,k).
E.g.f.: Series_Reversion( (1 - x) * (exp(x * (1 - x)) - 1) ). - Seiichi Manyama, Sep 09 2024

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

A371326 E.g.f. satisfies A(x) = log(1 + x/(1 - A(x)))/(1 - A(x))^2.

Original entry on oeis.org

0, 1, 5, 71, 1606, 50334, 2017840, 98597204, 5684225640, 377709287232, 28423701233784, 2389343434217376, 221907620769333648, 22565504728129558272, 2493614778861026071584, 297548320679718887153088, 38128996565367754662297600, 5222327925855459163424791680
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=1, n, (n+3*k-2)!/(n+2*k-1)!*stirling(n, k, 1));

Formula

a(n) = Sum_{k=1..n} (n+3*k-2)!/(n+2*k-1)! * Stirling1(n,k).
E.g.f.: Series_Reversion( (1 - x) * (exp(x * (1 - x)^2) - 1) ). - Seiichi Manyama, Sep 09 2024
Showing 1-3 of 3 results.