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.

A367474 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^2.

Original entry on oeis.org

1, 4, 28, 272, 3360, 50256, 881616, 17734944, 402278496, 10155145344, 282329361024, 8570500876032, 282047266728192, 10001430040080384, 380152962804068352, 15418451851593596928, 664633482628021493760, 30342827915683778027520
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Programs

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

Formula

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

A375945 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^(3/2).

Original entry on oeis.org

1, 3, 18, 156, 1758, 24342, 399480, 7577700, 163090500, 3926104860, 104520733560, 3048811591680, 96695722690200, 3312942954681240, 121938065727180480, 4798400761979259120, 201030443703421854480, 8933622147642363338160, 419725992843354254228640
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Programs

  • Mathematica
    nmax=18; CoefficientList[Series[1 / (1 + 2 * Log[1 - x])^(3/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a001147(n) = prod(k=0, n-1, 2*k+1);
    a(n) = sum(k=0, n, a001147(k+1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} A001147(k+1) * |Stirling1(n,k)|.
a(n) ~ n^(n+1) / (exp(n/2) * (exp(1/2) - 1)^(n + 3/2)). - Vaclav Kotesovec, Sep 06 2024

A375953 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^(5/2).

Original entry on oeis.org

1, 5, 40, 430, 5770, 92590, 1726940, 36682200, 873793620, 23061929940, 667868085360, 21052931727240, 717531427466280, 26289935772108120, 1030422613932910800, 43018144091244322560, 1905711682795871222160, 89284805444478025826640
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Programs

  • Mathematica
    nmax=17; CoefficientList[Series[1 / (1 + 2 * Log[1 - x])^(5/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a001147(n) = prod(k=0, n-1, 2*k+1);
    a(n) = sum(k=0, n, a001147(k+2)*abs(stirling(n, k, 1)))/3;

Formula

a(n) = (1/3) * Sum_{k=0..n} A001147(k+2) * |Stirling1(n,k)|.

A375722 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^3.

Original entry on oeis.org

1, 9, 117, 1962, 40122, 966276, 26755812, 836862192, 29167596504, 1120629465432, 47044646845848, 2142210019297680, 105154320625284240, 5534780654854980000, 310945503593770489440, 18570787974013838515200, 1174884522886771261079040
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-4 of 4 results.