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.

A352304 Expansion of e.g.f. 1/(exp(x) - x^4).

Original entry on oeis.org

1, -1, 1, -1, 25, -241, 1441, -6721, 67201, -1185409, 16652161, -180639361, 2098673281, -37526586241, 785718950017, -14516030954881, 247504017895681, -4832929862019841, 116556246644716801, -2930255897793414913, 69746855593499124481, -1673960044278244020481
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Cf. A089148, A352302, A352303.
Cf. A352300, A352311.

Programs

  • Mathematica
    m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^4), {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)))
  • 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.
a(n) ~ n! * (-1)^n / ((1 + LambertW(1/4)) * 2^(2*n + 10) * LambertW(1/4)^(n+4)). - Vaclav Kotesovec, Mar 12 2022
a(n) = n! * Sum_{k=0..floor(n/4)} (-k-1)^(n-4*k)/(n-4*k)!. - Seiichi Manyama, Aug 21 2024

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

Cf. A089148, A352310, A352311.
Cf. A352302, A352306.

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

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

Cf. A089148, A352309, A352311.
Cf. A352303.

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.

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