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.

A352307 Expansion of e.g.f. 1/(2 - exp(x) - x^3/6).

Original entry on oeis.org

1, 1, 3, 14, 83, 621, 5583, 58493, 700507, 9438253, 141291843, 2326680313, 41797029035, 813422096709, 17047913249279, 382815685896293, 9169316015977675, 233352842701661021, 6288004372005738747, 178851946015229702545, 5354894260179239755995
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 20; Range[0, m]! * CoefficientList[Series[1/(2 - 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/(2-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.

A352309 Expansion of e.g.f. 1/(exp(x) - x^2/2).

Original entry on oeis.org

1, -1, 2, -7, 31, -171, 1141, -8863, 78653, -785557, 8716861, -106395741, 1416724915, -20436548575, 317477947151, -5284248213091, 93816998697721, -1769737117839849, 35347571931577609, -745232024035027225, 16538641134235561631, -385387334950748244451
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 21; Range[0, m]! * CoefficientList[Series[1/(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/(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.
a(n) ~ n! * (-1)^n / ((1 + LambertW(1/sqrt(2))) * (2*LambertW(1/sqrt(2)))^(n+2)). - Vaclav Kotesovec, Mar 12 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (-k-1)^(n-2*k)/(2^k*(n-2*k)!). - Seiichi Manyama, Aug 21 2024

A352308 Expansion of e.g.f. 1/(2 - exp(x) - x^4/24).

Original entry on oeis.org

1, 1, 3, 13, 76, 551, 4803, 48833, 567465, 7418263, 107752293, 1721642143, 30008756055, 566650322031, 11523037802461, 251062618129063, 5834798259848815, 144078299659541361, 3766993649599221903, 103961442644871088897, 3020133228180079209075
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 20; Range[0, m]! * CoefficientList[Series[1/(2 - 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/(2-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.
Showing 1-3 of 3 results.