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.

A352303 Expansion of e.g.f. 1/(exp(x) - x^3).

Original entry on oeis.org

1, -1, 1, 5, -47, 239, -239, -11761, 170689, -1237825, -2360159, 238756319, -4146035519, 32586126143, 359988680689, -18567245926321, 351652342984321, -2283764958280321, -89760640709677247, 3866819337993369023, -74731210747948586879, 167887841949213912959
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^3), {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)))
    
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
    a(n) = b(n, 3);

Formula

a(n) = n * (n-1) * (n-2) * 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)/(n-3*k)!. - Seiichi Manyama, Aug 21 2024

A352300 Expansion of e.g.f. 1/(2 - exp(x) - x^4).

Original entry on oeis.org

1, 1, 3, 13, 99, 781, 7563, 84253, 1103595, 16074589, 260443083, 4630046653, 90017588235, 1894771249021, 42957132108075, 1043136555486493, 27024421701469995, 743851294350730141, 21679544916491784843, 666932347454809048189
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^4), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-x^4)))
    
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
    a(n) = b(n, 4);

Formula

a(n) = n * (n-1) * (n-2) * (n-3) * a(n-4) + Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 3.

A352302 Expansion of e.g.f. 1/(exp(x) - x^2).

Original entry on oeis.org

1, -1, 3, -13, 73, -521, 4441, -44185, 502545, -6429169, 91393201, -1429101521, 24378097129, -450504733849, 8965682806809, -191174795868841, 4348171177591201, -105077942935229537, 2688685949077138657, -72618903735812907553, 2064598911185525708601
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 20; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^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)))
    
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
    a(n) = b(n, 2);

Formula

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

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
Showing 1-4 of 4 results.