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 1-6 of 6 results.

A052820 Expansion of e.g.f. 1/(1 - x + log(1 - x)).

Original entry on oeis.org

1, 2, 9, 62, 572, 6604, 91526, 1480044, 27353448, 568731648, 13138994112, 333895239072, 9256507508112, 278000959058016, 8991458660924112, 311585506208924064, 11517363473843526912, 452332548042633835776
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Previous name was: A simple grammar.
a(n) is the number of ways to seat n people at circular tables, then linearly order the tables, then designate some (possibly all or none) of the tables at which only one person is seated. a(2) = 9 because we have: (1)(2), (1')(2), (1)(2'), (1')(2'), (2)(1), (2')(1), (2)(1'), (2')(1'), (1,2). Cf. A007840. - Geoffrey Critzer, Nov 05 2013

Crossrefs

Programs

  • Maple
    spec := [S,{C=Cycle(Z),B=Union(C,Z),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[1/(1-x+Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

Formula

E.g.f.: -1/(-1+x+log(-1/(-1+x))).
a(n) ~ n! * (1/(1-LambertW(1)))^n/(1/LambertW(1)-LambertW(1)). - Vaclav Kotesovec, Oct 01 2013
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

Extensions

New name using e.g.f., Vaclav Kotesovec, Oct 01 2013

A367845 Expansion of e.g.f. 1/(1 - x + log(1 - 2*x)).

Original entry on oeis.org

1, 3, 22, 250, 3816, 72968, 1675568, 44901456, 1375306368, 47392683648, 1814635323648, 76430014409472, 3511792144942080, 174806087920727040, 9370642040786049024, 538202280800536799232, 32972397141008692445184, 2146270648672407967137792
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 2^k * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ n! * 2^(n+1) / ((1/LambertW(1/(2*exp(1/2))) - 1 - 2*LambertW(1/(2*exp(1/2)))) * (1 - 2*LambertW(1/(2*exp(1/2))))^n). - Vaclav Kotesovec, Dec 02 2023

A367847 Expansion of e.g.f. 1/(1 - x + log(1 - 4*x)).

Original entry on oeis.org

1, 5, 66, 1358, 37592, 1304536, 54384080, 2646247152, 147186205056, 9210766696320, 640472632680192, 48989958019395840, 4087959251421060096, 369547591764702870528, 35976590549993421907968, 3752609987262290143082496, 417518648351593243448279040
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 4^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 4^k * (k-1)! * binomial(n,k) * a(n-k).

A367852 Expansion of e.g.f. 1/(1 - x + log(1 - 3*x)/3).

Original entry on oeis.org

1, 2, 11, 102, 1320, 21804, 436986, 10283580, 277697304, 8458929792, 286825214592, 10712216384352, 436859348261904, 19313926491051360, 920053448561989296, 46977842202096405024, 2559387620091962391552, 148187802162935002975488
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 3^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

A367923 Expansion of e.g.f. 1/(1 - x + 3*log(1 - x)).

Original entry on oeis.org

1, 4, 35, 462, 8136, 179112, 4731786, 145838844, 5137045848, 203566459392, 8963064065088, 434109674396736, 22936702911358608, 1312878755037640320, 80928769156102447920, 5344960170283958863008, 376543135663291116638208, 28184733661095459402610176
Offset: 0

Views

Author

Seiichi Manyama, Dec 05 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+3*sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + 3 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).

A367829 E.g.f. A(x) satisfies A(x) = (1 - log(1 - x) * A(3*x)) / (1 - x).

Original entry on oeis.org

1, 2, 17, 530, 60332, 24882484, 36501847110, 186651759218364, 3267898148335418280, 193010228785740170125728, 37993098362777240856612204096, 24678625994736515097158433120107040, 52461378922253347510159057679901573120528
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^(i-j)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 3^(n-k) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-6 of 6 results.