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.

A352306 Expansion of e.g.f. 1/(2 - exp(x) - x^2/2).

Original entry on oeis.org

1, 1, 4, 19, 129, 1071, 10743, 125455, 1675439, 25167073, 420070323, 7712503173, 154475622513, 3351859639363, 78324320723561, 1960968388497523, 52368881358012435, 1485952518531483045, 44643697199669589447, 1415782273405809697009
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 19; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - x^2/2), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-x^2/2)))
    
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (1+(k==m))*binomial(n, k)*b(n-k, m)));
    a(n) = b(n, 2);

Formula

a(n) = binomial(n,2) * a(n-2) + Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 1.

A352311 Expansion of e.g.f. 1/(exp(x) - x^4/24).

Original entry on oeis.org

1, -1, 1, -1, 2, -11, 61, -281, 1191, -5923, 41791, -354091, 2968021, -24059751, 204718515, -1996937671, 22125450621, -258434553861, 3056858429581, -37181421375349, 482010195953821, -6741275765687821, 99663246605243861, -1521712424934601901
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 23; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^4/24), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^4/24)))
    
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m))*binomial(n, k)*b(n-k, m)));
    a(n) = b(n, 4);

Formula

a(n) = binomial(n,4) * a(n-4) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 3.
a(n) ~ n! * 3*(-1)^n / ((1 + LambertW(3^(1/4) / 2^(5/4))) * 2^(2*n + 7) * LambertW(3^(1/4) / 2^(5/4))^(n+4)). - Vaclav Kotesovec, Mar 12 2022
a(n) = n! * Sum_{k=0..floor(n/4)} (-k-1)^(n-4*k)/(24^k*(n-4*k)!). - Seiichi Manyama, Aug 21 2024

A352310 Expansion of e.g.f. 1/(exp(x) - x^3/6).

Original entry on oeis.org

1, -1, 1, 0, -7, 39, -139, 139, 3249, -38305, 257641, -724681, -9925519, 208718223, -2209932451, 11619569779, 98841199521, -3691083087521, 56488651405393, -466578080641297, -1989509977776479, 159427986446212959, -3372599255892634459, 39809520784433784075
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 23; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^3/6), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^3/6)))
    
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m))*binomial(n, k)*b(n-k, m)));
    a(n) = b(n, 3);

Formula

a(n) = binomial(n,3) * a(n-3) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 2.
a(n) = n! * Sum_{k=0..floor(n/3)} (-k-1)^(n-3*k)/(6^k*(n-3*k)!). - Seiichi Manyama, Aug 21 2024
Showing 1-3 of 3 results.