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.

A371006 Expansion of e.g.f. (1/x) * Series_Reversion( x/(3*exp(x) - 2) ).

Original entry on oeis.org

1, 3, 21, 246, 4143, 91938, 2543457, 84476766, 3278575515, 145703001450, 7299102908613, 407061606983430, 25016221521245703, 1679926053870309378, 122399565517464024009, 9617404242454811783598, 810684382032520533507891, 72976185712308646408856538
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = 1/(n+1) * Sum_{k=0..n+1} 3^k * (-2)^(n+1-k) * k^n * binomial(n+1,k).
a(n) = n! * Sum_{k=0..n} 3^k * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024

A371007 Expansion of e.g.f. (1/x) * Series_Reversion( 2*x/(3*exp(2*x) - 1) ).

Original entry on oeis.org

1, 3, 24, 336, 6864, 185808, 6286560, 255703584, 12163234560, 662866302720, 40735968170496, 2787616114300416, 210253334027606016, 17331011952028981248, 1550159522438672412672, 149539908497083261980672, 15476976326308703371984896
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2024

Keywords

Crossrefs

Cf. A371005.

Programs

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

Formula

a(n) = 1/(2*(n+1)) * Sum_{k=0..n+1} 3^k * (-1)^(n+1-k) * k^n * binomial(n+1,k).
a(n) = n! * Sum_{k=0..n} 3^k * 2^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024

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)!.
Showing 1-3 of 3 results.