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.

A360482 E.g.f. satisfies A(x) = x * exp(x + 3 * A(x)).

Original entry on oeis.org

0, 1, 8, 120, 2848, 92960, 3868224, 195810496, 11680512512, 802445898240, 62396469222400, 5417515922441216, 519519435065020416, 54535504354085687296, 6219954774471102242816, 765903524713482618101760, 101269330068289021683564544, 14310318526812295078276628480
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).
a(n) ~ sqrt(1 + LambertW(exp(-1)/3)) * n^(n-1) /(3 * exp(n) * LambertW(exp(-1)/3)^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).

A360471 E.g.f. satisfies A(x) = x * exp( 2*A(x) + x * exp(2*A(x)) ).

Original entry on oeis.org

0, 1, 6, 75, 1476, 39805, 1366278, 56998179, 2800588808, 158420939193, 10140538486410, 724652822705119, 57187947315670284, 4939834587311520117, 463572330418586227790, 46965096302630022564315, 5108915146530700018466832, 593925863391217441843199089
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=1, n, k^(n-k)*(2*n)^(k-1)*binomial(n, k));

Formula

a(n) = Sum_{k=1..n} k^(n-k) * (2*n)^(k-1) * binomial(n,k).
a(n) ~ 2^(n - 1/2) * s^n * n^(n-1) / (sqrt(2 + 1/s - 4*s) * (1 - 2*s)^n * exp(n*(1 - 2*s))), where s = 0.3875920123187127910093095185777835252050660050582... is the root of the equation 2*s*(1 + LambertW(s)) = 1. - Vaclav Kotesovec, Feb 17 2023
Showing 1-4 of 4 results.