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.

A352299 Expansion of e.g.f. 1/(2 - exp(x) - x^3).

Original entry on oeis.org

1, 1, 3, 19, 123, 1021, 10683, 127093, 1725867, 26535613, 452307243, 8475606613, 173390108235, 3842119808749, 91675559886459, 2343875745873493, 63920729617231275, 1852126733351677021, 56823327291638414667, 1840195730889731550805
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^3), {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^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.

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

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.
Showing 1-4 of 4 results.