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

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

Cf. A006155, A346269, A352300.
Cf. A352303, A352307.

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.

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

Cf. A006155, A352306, A352307.
Cf. A352300.

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.

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