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.

A377789 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - 2*log(1-x)) ).

Original entry on oeis.org

1, 2, 10, 88, 1148, 20088, 442896, 11802096, 369132256, 13261156416, 538227938880, 24359100451200, 1216403663398656, 66440221207025664, 3940468338389603328, 252190997066643909120, 17324237625466992906240, 1271459220768570290626560, 99289436336361780797288448
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} 2^k * |Stirling1(n,k)|/(n-k+1)!.

A370940 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - 2*log(1+x)) ).

Original entry on oeis.org

1, 2, 14, 184, 3612, 94968, 3139088, 125181936, 5851551680, 313874206656, 19006905318528, 1282738818650496, 95477483835672960, 7770589670409684480, 686519279618695022592, 65436589709543394150912, 6693486627002144059422720, 731378220534326743907266560
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^k * (n+k)! * Stirling1(n,k).
a(n) ~ LambertW(exp(1/2))^n * n^(n-1) / (sqrt(1 + LambertW(exp(1/2))) * 2^(n+1) * exp(n) * (1 - LambertW(exp(1/2)))^(2*n+1)). - Vaclav Kotesovec, Mar 06 2024

A371297 E.g.f. satisfies A(x) = 1/(1 + 2*log(1 - x*A(x)^2)).

Original entry on oeis.org

1, 2, 26, 676, 26852, 1443888, 98183024, 8083614880, 781958648448, 86940057459840, 10925288128027968, 1531414930604605440, 236905910564035082112, 40093453025252047368192, 7368774639911257328778240, 1461607086204159742139338752, 311206233406111454756938844160
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, 2^k*(2*n+k)!*abs(stirling(n, k, 1)))/(2*n+1)!;

Formula

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} 2^k * (2*n+k)! * |Stirling1(n,k)|.
Showing 1-3 of 3 results.