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.

A351933 Expansion of e.g.f. exp(x / (1 - x^2/2)).

Original entry on oeis.org

1, 1, 1, 4, 13, 61, 331, 1996, 14449, 109873, 971821, 8995636, 93329941, 1018571269, 12110589583, 151955795356, 2037757374241, 28837620752161, 430834395468889, 6777014821152868, 111663525724783741, 1930478057636642221, 34781068833200111731
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A000262, A351934, A351935, A351936.
Cf. A088009.

Programs

  • Mathematica
    m = 22; Range[0, m]! * CoefficientList[Series[Exp[x / (1 - x^2/2)], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^2/2))))
  • PARI
    a(n) = if(n<3, 1, sum(k=0, (n-1)\2, (2*k+1)!/2^k*binomial(n-1, 2*k)*a(n-1-2*k)));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (2*k+1)!/2^k * binomial(n-1,2*k) * a(n-1-2*k) for n > 2.
a(n) ~ n^(n - 1/4) / (2^(n/2 + 5/8) * exp(n - 2^(3/4)*sqrt(n))). - Vaclav Kotesovec, Mar 03 2022
Conjecture D-finite with recurrence +4*a(n) -4*a(n-1) -4*(n-1)*(n-2)*a(n-2) -2*(n-1)*(n-2)*a(n-3) +(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Mar 09 2022
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-k-1,k)/(2^k * (n-2*k)!). - Seiichi Manyama, Jun 08 2024

A351934 Expansion of e.g.f. exp(x / (1 - x^3/6)).

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 281, 1961, 10585, 59641, 509521, 4086061, 29792621, 288051765, 2997900361, 27857436881, 302238265841, 3759751338481, 43054257392545, 526471319508181, 7463835257515141, 101755979383832621, 1404876681654178041, 22115830210157954425
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A000262, A351933, A351935, A351936.
Cf. A293493.

Programs

  • Mathematica
    m = 24; Range[0, m]! * CoefficientList[Series[Exp[x/(1 - x^3/6)], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^3/6))))
  • PARI
    a(n) = if(n<4, 1, sum(k=0, (n-1)\3, (3*k+1)!/6^k*binomial(n-1, 3*k)*a(n-1-3*k)));

Formula

a(n) = Sum_{k=0..floor((n-1)/3)} (3*k+1)!/6^k * binomial(n-1,3*k) * a(n-1-3*k) for n > 3.
a(n) ~ n^(n - 1/4) / (2^(n/3 + 5/12) * 3^(n/3 + 1/6) * exp(n - 2^(7/6) * 3^(-1/3) * sqrt(n) - 6^(-2/3))). - Vaclav Kotesovec, Mar 03 2022
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k)/(6^k * (n-3*k)!). - Seiichi Manyama, Jun 08 2024

A351936 Expansion of e.g.f. exp(x / (1 - x^5/5!)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 43, 169, 505, 1261, 5545, 55441, 442729, 2540539, 11381371, 54534481, 548974609, 6572212921, 59711454433, 413207026561, 2551872368305, 24405087826351, 356232375255835, 4526838244526137, 44179554690486601, 358234717042702501
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A000262, A351933, A351934, A351935.

Programs

  • Mathematica
    m = 26; Range[0, m]! * CoefficientList[Series[Exp[x/(1 - x^5/5!)], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^5/5!))))
  • PARI
    a(n) = if(n<6, 1, sum(k=0, (n-1)\5, (5*k+1)!/5!^k*binomial(n-1, 5*k)*a(n-1-5*k)));

Formula

a(n) = Sum_{k=0..floor((n-1)/5)} (5*k+1)!/(5!)^k * binomial(n-1,5*k) * a(n-1-5*k) for n > 5.
a(n) = n! * Sum_{k=0..floor(n/5)} binomial(n-4*k-1,k)/(120^k * (n-5*k)!). - Seiichi Manyama, Jun 08 2024
Showing 1-3 of 3 results.

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