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.

A360481 E.g.f. satisfies A(x) = x * exp(x + 2 * A(x)).

Original entry on oeis.org

0, 1, 6, 63, 1044, 23805, 692118, 24482115, 1020584232, 49000005945, 2662853279850, 161586078510879, 10830019921469532, 794577001293803637, 63339899145968483262, 5451312770064188283195, 503784284643602483767632, 49757423537114340032969073
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(x))/2)))
    
  • PARI
    a(n) = sum(k=1, n, 2^(k-1)*k^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = -LambertW(-2*x * exp(x))/2.
a(n) = Sum_{k=1..n} 2^(k-1) * k^(n-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1)/2)) * n^(n-1) / (2 * LambertW(exp(-1)/2)^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023

A360483 E.g.f. satisfies A(x) = x * exp(x - 2 * A(x)).

Original entry on oeis.org

0, 1, -2, 15, -172, 2685, -53226, 1281091, -36296408, 1183527225, -43660076950, 1797823266591, -81746462498724, 4068086310006901, -219929012455113794, 12835335232410655035, -804287930238495495856, 53858337558670992931185
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(lambertw(2*x*exp(x))/2)))
    
  • PARI
    a(n) = sum(k=1, n, (-2)^(k-1)*k^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = LambertW(2*x * exp(x))/2.
a(n) = Sum_{k=1..n} (-2)^(k-1) * k^(n-1) * binomial(n,k).

A360484 E.g.f. satisfies A(x) = x * exp(x - 3 * A(x)).

Original entry on oeis.org

0, 1, -4, 48, -896, 22880, -743232, 29337280, -1363752448, 72979407360, -4419108684800, 298730433250304, -22300928914403328, 1822195561572585472, -161756111552270491648, 15501595224386724126720, -1595092357302221461127168, 175405731698165304882495488
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(lambertw(3*x*exp(x))/3)))
    
  • PARI
    a(n) = sum(k=1, n, (-3)^(k-1)*k^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = LambertW(3*x * exp(x))/3.
a(n) = Sum_{k=1..n} (-3)^(k-1) * k^(n-1) * binomial(n,k).

A360545 E.g.f. satisfies A(x) = x * exp( 3*(x + A(x))/2 ).

Original entry on oeis.org

0, 1, 6, 54, 756, 14580, 358668, 10736712, 378823392, 15395255280, 708217959600, 36380741745744, 2064234271203360, 128214974795177088, 8652900673357097472, 630483717450225530880, 49330027417316557012992, 4124992361928178722764544
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-2*lambertw(-3*x/2*exp(3*x/2))/3)))
    
  • PARI
    a(n) = sum(k=1, n, (3*k/2)^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = (-2/3) * LambertW(-3*x/2 * exp(3*x/2)).
a(n) = Sum_{k=1..n} (3*k/2)^(n-1) * binomial(n,k) = 3^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023
Showing 1-4 of 4 results.